Method and apparatus for video coding

ABSTRACT

Aspects of the disclosure provide methods and apparatuses for video encoding/decoding. In some examples, an apparatus for video decoding includes processing circuitry. The processing circuitry can decode coding information of a coefficient block from a coded video bitstream. The coding information can indicate a size of the coefficient block. The processing circuitry can determine, based on the size of the coefficient block, an order in which inverse horizontal and inverse vertical transforms of an inverse primary transform are to be performed on transform coefficients of the coefficient block to obtain residual data of a residual block. When the size of the coefficient block satisfies a condition, the inverse vertical transform is performed after the inverse horizontal transform is performed on the transform coefficients of the coefficient block. The processing circuitry can reconstruct a sample in the residual block based on the residual data.

INCORPORATION BY REFERENCE

This present application claims the benefit of priority to U.S.Provisional Application No. 62/816,124, “Adaptive Transform CoefficientZero-Out” filed on Mar. 9, 2019, which is incorporated by referenceherein in its entirety.

TECHNICAL FIELD

The present disclosure describes embodiments generally related to videocoding.

BACKGROUND

The background description provided herein is for the purpose ofgenerally presenting the context of the disclosure. Work of thepresently named inventors, to the extent the work is described in thisbackground section, as well as aspects of the description that may nototherwise qualify as prior art at the time of filing, are neitherexpressly nor impliedly admitted as prior art against the presentdisclosure.

Video coding and decoding can be performed using inter-pictureprediction with motion compensation. Uncompressed digital video caninclude a series of pictures, each picture having a spatial dimensionof, for example, 1920×1080 luminance samples and associated chrominancesamples. The series of pictures can have a fixed or variable picturerate (informally also known as frame rate), of, for example 60 picturesper second or 60 Hz. Uncompressed video has significant bitraterequirements. For example, 1080p60 4:2:0 video at 8 bit per sample(1920×1080 luminance sample resolution at 60 Hz frame rate) requiresclose to 1.5 Gbit/s bandwidth. An hour of such video requires more than600 GBytes of storage space.

One purpose of video coding and decoding can be the reduction ofredundancy in the input video signal, through compression. Compressioncan help reduce the aforementioned bandwidth or storage spacerequirements, in some cases by two orders of magnitude or more. Bothlossless and lossy compression, as well as a combination thereof can beemployed. Lossless compression refers to techniques where an exact copyof the original signal can be reconstructed from the compressed originalsignal. When using lossy compression, the reconstructed signal may notbe identical to the original signal, but the distortion between originaland reconstructed signals is small enough to make the reconstructedsignal useful for the intended application. In the case of video, lossycompression is widely employed. The amount of distortion tolerateddepends on the application; for example, users of certain consumerstreaming applications may tolerate higher distortion than users oftelevision distribution applications. The compression ratio achievablecan reflect that: higher allowable/tolerable distortion can yield highercompression ratios.

A video encoder and decoder can utilize techniques from several broadcategories, including, for example, motion compensation, transform,quantization, and entropy coding.

Video codec technologies can include techniques known as intra coding.In intra coding, sample values are represented without reference tosamples or other data from previously reconstructed reference pictures.In some video codecs, the picture is spatially subdivided into blocks ofsamples. When all blocks of samples are coded in intra mode, thatpicture can be an intra picture. Intra pictures and their derivationssuch as independent decoder refresh pictures, can be used to reset thedecoder state and can, therefore, be used as the first picture in acoded video bitstream and a video session, or as a still image. Thesamples of an intra block can be exposed to a transform, and thetransform coefficients can be quantized before entropy coding. Intraprediction can be a technique that minimizes sample values in thepre-transform domain. In some cases, the smaller the DC value after atransform is, and the smaller the AC coefficients are, the fewer thebits that are required at a given quantization step size to representthe block after entropy coding.

Traditional intra coding such as known from, for example MPEG-2generation coding technologies, does not use intra prediction. However,some newer video compression technologies include techniques thatattempt, from, for example, surrounding sample data and/or metadataobtained during the encoding/decoding of spatially neighboring, andpreceding in decoding order, blocks of data. Such techniques arehenceforth called “intra prediction” techniques. Note that in at leastsome cases, intra prediction is only using reference data from thecurrent picture under reconstruction and not from reference pictures.

There can be many different forms of intra prediction. When more thanone of such techniques can be used in a given video coding technology,the technique in use can be coded in an intra prediction mode. Incertain cases, modes can have submodes and/or parameters, and those canbe coded individually or included in the mode codeword. Which codewordto use for a given mode/submode/parameter combination can have an impactin the coding efficiency gain through intra prediction, and so can theentropy coding technology used to translate the codewords into abitstream.

A certain mode of intra prediction was introduced with H.264, refined inH.265, and further refined in newer coding technologies such as jointexploration model (JEM), versatile video coding (VVC), and benchmark set(BMS). A predictor block can be formed using neighboring sample valuesbelonging to already available samples. Sample values of neighboringsamples are copied into the predictor block according to a direction. Areference to the direction in use can be coded in the bitstream or mayitself be predicted.

Referring to FIG. 1A, depicted in the lower right is a subset of ninepredictor directions known from H.265's 33 possible predictor directions(corresponding to the 33 angular modes of the 35 intra modes). The pointwhere the arrows converge (101) represents the sample being predicted.The arrows represent the direction from which the sample is beingpredicted. For example, arrow (102) indicates that sample (101) ispredicted from a sample or samples to the upper right, at a 45 degreeangle from the horizontal. Similarly, arrow (103) indicates that sample(101) is predicted from a sample or samples to the lower left of sample(101), in a 22.5 degree angle from the horizontal.

Still referring to FIG. 1A, on the top left there is depicted a squareblock (104) of 4×4 samples (indicated by a dashed, boldface line). Thesquare block (104) includes 16 samples, each labelled with an “S”, itsposition in the Y dimension (e.g., row index) and its position in the Xdimension (e.g., column index). For example, sample S21 is the secondsample in the Y dimension (from the top) and the first (from the left)sample in the X dimension. Similarly, sample S44 is the fourth sample inblock (104) in both the Y and X dimensions. As the block is 4×4 samplesin size, S44 is at the bottom right. Further shown are reference samplesthat follow a similar numbering scheme. A reference sample is labelledwith an R, its Y position (e.g., row index) and X position (columnindex) relative to block (104). In both H.264 and H.265, predictionsamples neighbor the block under reconstruction; therefore no negativevalues need to be used.

Intra picture prediction can work by copying reference sample valuesfrom the neighboring samples as appropriated by the signaled predictiondirection. For example, assume the coded video bitstream includessignaling that, for this block, indicates a prediction directionconsistent with arrow (102)—that is, samples are predicted from aprediction sample or samples to the upper right, at a 45 degree anglefrom the horizontal. In that case, samples S41, S32, S23, and S14 arepredicted from the same reference sample R05. Sample S44 is thenpredicted from reference sample R08.

In certain cases, the values of multiple reference samples may becombined, for example through interpolation, in order to calculate areference sample; especially when the directions are not evenlydivisible by 45 degrees.

The number of possible directions has increased as video codingtechnology has developed. In H.264 (year 2003), nine different directioncould be represented. That increased to 33 in H.265 (year 2013), andJEMNVC/BMS, at the time of disclosure, can support up to 65 directions.Experiments have been conducted to identify the most likely directions,and certain techniques in the entropy coding are used to represent thoselikely directions in a small number of bits, accepting a certain penaltyfor less likely directions. Further, the directions themselves cansometimes be predicted from neighboring directions used in neighboring,already decoded, blocks.

FIG. 1B shows a schematic (180) that depicts 65 intra predictiondirections according to JEM to illustrate the increasing number ofprediction directions over time.

The mapping of intra prediction directions bits in the coded videobitstream that represent the direction can be different from videocoding technology to video coding technology; and can range, forexample, from simple direct mappings of prediction direction to intraprediction mode, to codewords, to complex adaptive schemes involvingmost probable modes, and similar techniques. In all cases, however,there can be certain directions that are statistically less likely tooccur in video content than certain other directions. As the goal ofvideo compression is the reduction of redundancy, those less likelydirections will, in a well working video coding technology, berepresented by a larger number of bits than more likely directions.

Video coding and decoding can be performed using inter-pictureprediction with motion compensation. Motion compensation can be a lossycompression technique and can relate to techniques where a block ofsample data from a previously reconstructed picture or part thereof(reference picture), after being spatially shifted in a directionindicated by a motion vector (MV henceforth), is used for the predictionof a newly reconstructed picture or picture part. In some cases, thereference picture can be the same as the picture currently underreconstruction. MVs can have two dimensions X and Y, or threedimensions, the third being an indication of the reference picture inuse (the latter, indirectly, can be a time dimension).

In some video compression techniques, an MV applicable to a certain areaof sample data can be predicted from other MVs, for example from thoserelated to another area of sample data spatially adjacent to the areaunder reconstruction, and preceding that MV in decoding order. Doing socan substantially reduce the amount of data required for coding the MV,thereby removing redundancy and increasing compression. MV predictioncan work effectively, for example, because when coding an input videosignal derived from a camera (known as natural video) there is astatistical likelihood that areas larger than the area to which a singleMV is applicable move in a similar direction and, therefore, can in somecases be predicted using a similar motion vector derived from MVs ofneighboring area. That results in the MV found for a given area to besimilar or the same as the MV predicted from the surrounding MVs, andthat in turn can be represented, after entropy coding, in a smallernumber of bits than what would be used if coding the MV directly. Insome cases, MV prediction can be an example of lossless compression of asignal (namely: the MVs) derived from the original signal (namely: thesample stream). In other cases, MV prediction itself can be lossy, forexample because of rounding errors when calculating a predictor fromseveral surrounding MVs.

Various MV prediction mechanisms are described in H.265/HEVC (ITU-T Rec.H.265, “High Efficiency Video Coding”, December 2016). Out of the manyMV prediction mechanisms that H.265 offers, described here is atechnique henceforth referred to as “spatial merge”.

Referring to FIG. 2, a current block (201) comprises samples that havebeen found by the encoder during the motion search process to bepredictable from a previous block of the same size that has beenspatially shifted. Instead of coding that MV directly, the MV can bederived from metadata associated with one or more reference pictures,for example from the most recent (in decoding order) reference picture,using the MV associated with either one of five surrounding samples,denoted A0, A1, and B0, B1, B2 (202 through 206, respectively). InH.265, the MV prediction can use predictors from the same referencepicture that the neighboring block is using.

SUMMARY

Aspects of the disclosure provide methods and apparatuses for videoencoding/decoding. In some examples, an apparatus for video decodingincludes processing circuitry. The processing circuitry can decodecoding information of a coefficient block from a coded video bitstream.The coding information can indicate a size of the coefficient block. Theprocessing circuitry can determine, based on the size of the coefficientblock, an order in which inverse horizontal and inverse verticaltransforms of an inverse primary transform are to be performed ontransform coefficients of the coefficient block to obtain residual dataof a residual block. When the size of the coefficient block satisfies acondition, the inverse vertical transform is performed after the inversehorizontal transform is performed on the transform coefficients of thecoefficient block. The processing circuitry can reconstruct a sample inthe residual block based on the residual data. In an example, thecondition is that the size of the coefficient block is 32×64. In anexample, the condition is that a height N of the coefficient block ofM×N is larger than a width M of the coefficient block. In an example,the size of the coefficient block is M×N, where M and N are positiveintegers. First residual data in a m×n region in the residual block areto be calculated by the inverse primary transform and second residualdata outside the m×n region in the residual block are not to becalculated by the inverse primary transform where m is less than orequal to M, and n is less than or equal to N. The condition is that aratio m/M is larger than or equal to a ratio n/N.

Aspects of the disclosure provide methods and apparatuses for videoencoding/decoding. In some examples, an apparatus for video decodingincludes processing circuitry. The processing circuitry can decodecoding information of a coefficient block from a coded video bitstream.The coding information can indicate a size of the coefficient block. Theprocessing circuitry can determine, based on the size of the coefficientblock, whether to reduce a number of calculations in one of inversehorizontal and inverse vertical transforms of an inverse primarytransform. The inverse vertical transform can transform coefficients ofthe coefficient block to intermediate data of an intermediate block, andthe inverse horizontal transform can transform the intermediate data toresidual data in a residual block. The processing circuitry can performthe inverse primary transform. When the number of calculations in theinverse vertical transform is determined to be reduced, top 16 rows ofthe intermediate data in the intermediate block are calculated by theinverse vertical transform and the remaining intermediate data in theintermediate block are zero. When the number of calculations in theinverse horizontal transform is determined to be reduced, left 16columns of the residual data in the residual block are calculated by theinverse horizontal transform and the remaining residual data in theresidual block are zero. The processing circuitry can reconstruct asample in the residual block based on the residual data.

In an embodiment, the size of the coefficient block is 32×64. The one ofthe inverse horizontal and inverse vertical transforms is the inversevertical transform. The processing circuitry can determine that thenumber of calculations in the inverse vertical transform is to bereduced when the size of the coefficient block is 32×64. The processingcircuitry can perform the inverse primary transform where the top 16rows of the intermediate data in the intermediate block can becalculated by the inverse vertical transform and the remainingintermediate data in the intermediate block are zero.

In an embodiment, the size of the coefficient block is 32×64. The one ofthe inverse horizontal and inverse vertical transforms is the inversehorizontal transform. The processing circuitry can determine that thenumber of calculations in the inverse horizontal transform is to bereduced when the size of the coefficient block is 32×64. The processingcircuitry can perform the inverse primary transform where the left 16columns of the residual data in the residual block can be calculated bythe inverse horizontal transform and the remaining residual data in theresidual block are zero.

In an embodiment, the size of the coefficient block is 32×32. The one ofthe inverse horizontal and inverse vertical transforms is the inversehorizontal transform. The processing circuitry can determine that thenumber of calculations in the inverse horizontal transform is to bereduced when the size of the coefficient block is 32×32. The processingcircuitry can perform the inverse primary transform where the left 16columns of the residual data in the residual block can be calculated bythe inverse horizontal transform, the remaining residual data in theresidual block are zero, and the intermediate data in the intermediateblock can be calculated by the inverse vertical transform.

In an embodiment, the size of the coefficient block is 32×32. The one ofthe inverse horizontal and inverse vertical transforms is the inversevertical transform. The processing circuitry can determine that thenumber of calculations in the inverse vertical transform is to bereduced when the size of the coefficient block is 32×32. The processingcircuitry can perform the inverse primary transform where the top 16rows of the intermediate data in the intermediate block can becalculated by the inverse vertical transform, the remaining intermediatedata in the intermediate block are zero, and the residual data in theresidual block can be calculated by the inverse horizontal transform.

BRIEF DESCRIPTION OF THE DRAWINGS

Further features, the nature, and various advantages of the disclosedsubject matter will be more apparent from the following detaileddescription and the accompanying drawings in which:

FIG. 1A is a schematic illustration of an exemplary subset of intraprediction modes.

FIG. 1B is an illustration of exemplary intra prediction directions.

FIG. 2 is a schematic illustration of a current block and itssurrounding spatial merge candidates in one example.

FIG. 3 is a schematic illustration of a simplified block diagram of acommunication system (300) in accordance with an embodiment.

FIG. 4 is a schematic illustration of a simplified block diagram of acommunication system (400) in accordance with an embodiment.

FIG. 5 is a schematic illustration of a simplified block diagram of adecoder in accordance with an embodiment.

FIG. 6 is a schematic illustration of a simplified block diagram of anencoder in accordance with an embodiment.

FIG. 7 shows a block diagram of an encoder in accordance with anotherembodiment.

FIG. 8 shows a block diagram of a decoder in accordance with anotherembodiment.

FIG. 9 shows an example of transform unit syntax in accordance with anembodiment.

FIGS. 10A-10C show an example of residual coding syntax in accordancewith an embodiment.

FIGS. 11A-11B show examples of primary transforms in accordance with anembodiment.

FIGS. 12A-12E show an exemplary transformation process in accordancewith an embodiment.

FIG. 13 shows an example of residual coding syntax in accordance withanother embodiment.

FIG. 14 shows a flow chart outlining a process (1400) in accordance withan embodiment.

FIG. 15 shows a flow chart outlining a process (1500) in accordance withan embodiment.

FIG. 16 is a schematic illustration of a computer system in accordancewith an embodiment.

DETAILED DESCRIPTION OF EMBODIMENTS

FIG. 3 illustrates a simplified block diagram of a communication system(300) according to an embodiment of the present disclosure. Thecommunication system (300) includes a plurality of terminal devices thatcan communicate with each other, via, for example, a network (350). Forexample, the communication system (300) includes a first pair ofterminal devices (310) and (320) interconnected via the network (350).In the FIG. 3 example, the first pair of terminal devices (310) and(320) performs unidirectional transmission of data. For example, theterminal device (310) may code video data (e.g., a stream of videopictures that are captured by the terminal device (310)) fortransmission to the other terminal device (320) via the network (350).The encoded video data can be transmitted in the form of one or morecoded video bitstreams. The terminal device (320) may receive the codedvideo data from the network (350), decode the coded video data torecover the video pictures and display video pictures according to therecovered video data. Unidirectional data transmission may be common inmedia serving applications and the like.

In another example, the communication system (300) includes a secondpair of terminal devices (330) and (340) that performs bidirectionaltransmission of coded video data that may occur, for example, duringvideoconferencing. For bidirectional transmission of data, in anexample, each terminal device of the terminal devices (330) and (340)may code video data (e.g., a stream of video pictures that are capturedby the terminal device) for transmission to the other terminal device ofthe terminal devices (330) and (340) via the network (350). Eachterminal device of the terminal devices (330) and (340) also may receivethe coded video data transmitted by the other terminal device of theterminal devices (330) and (340), and may decode the coded video data torecover the video pictures and may display video pictures at anaccessible display device according to the recovered video data.

In the FIG. 3 example, the terminal devices (310), (320), (330) and(340) may be illustrated as servers, personal computers and smart phonesbut the principles of the present disclosure may be not so limited.Embodiments of the present disclosure find application with laptopcomputers, tablet computers, media players and/or dedicated videoconferencing equipment. The network (350) represents any number ofnetworks that convey coded video data among the terminal devices (310),(320), (330) and (340), including for example wireline (wired) and/orwireless communication networks. The communication network (350) mayexchange data in circuit-switched and/or packet-switched channels.Representative networks include telecommunications networks, local areanetworks, wide area networks and/or the Internet. For the purposes ofthe present discussion, the architecture and topology of the network(350) may be immaterial to the operation of the present disclosureunless explained herein below.

FIG. 4 illustrates, as an example for an application for the disclosedsubject matter, the placement of a video encoder and a video decoder ina streaming environment. The disclosed subject matter can be equallyapplicable to other video enabled applications, including, for example,video conferencing, digital TV, storing of compressed video on digitalmedia including CD, DVD, memory stick and the like, and so on.

A streaming system may include a capture subsystem (413), that caninclude a video source (401), for example a digital camera, creating forexample a stream of video pictures (402) that are uncompressed. In anexample, the stream of video pictures (402) includes samples that aretaken by the digital camera. The stream of video pictures (402),depicted as a bold line to emphasize a high data volume when compared toencoded video data (404) (or coded video bitstreams), can be processedby an electronic device (420) that includes a video encoder (403)coupled to the video source (401). The video encoder (403) can includehardware, software, or a combination thereof to enable or implementaspects of the disclosed subject matter as described in more detailbelow. The encoded video data (404) (or encoded video bitstream (404)),depicted as a thin line to emphasize the lower data volume when comparedto the stream of video pictures (402), can be stored on a streamingserver (405) for future use. One or more streaming client subsystems,such as client subsystems (406) and (408) in FIG. 4 can access thestreaming server (405) to retrieve copies (407) and (409) of the encodedvideo data (404). A client subsystem (406) can include a video decoder(410), for example, in an electronic device (430). The video decoder(410) decodes the incoming copy (407) of the encoded video data andcreates an outgoing stream of video pictures (411) that can be renderedon a display (412) (e.g., display screen) or other rendering device (notdepicted). In some streaming systems, the encoded video data (404),(407), and (409) (e.g., video bitstreams) can be encoded according tocertain video coding/compression standards. Examples of those standardsinclude ITU-T Recommendation H.265. In an example, a video codingstandard under development is informally known as Versatile Video Coding(VVC). The disclosed subject matter may be used in the context of VVC.

It is noted that the electronic devices (420) and (430) can includeother components (not shown). For example, the electronic device (420)can include a video decoder (not shown) and the electronic device (430)can include a video encoder (not shown) as well.

FIG. 5 shows a block diagram of a video decoder (510) according to anembodiment of the present disclosure. The video decoder (510) can beincluded in an electronic device (530). The electronic device (530) caninclude a receiver (531) (e.g., receiving circuitry). The video decoder(510) can be used in the place of the video decoder (410) in the FIG. 4example.

The receiver (531) may receive one or more coded video sequences to bedecoded by the video decoder (510); in the same or another embodiment,one coded video sequence at a time, where the decoding of each codedvideo sequence is independent from other coded video sequences. Thecoded video sequence may be received from a channel (501), which may bea hardware/software link to a storage device which stores the encodedvideo data. The receiver (531) may receive the encoded video data withother data, for example, coded audio data and/or ancillary data streams,that may be forwarded to their respective using entities (not depicted).The receiver (531) may separate the coded video sequence from the otherdata. To combat network jitter, a buffer memory (515) may be coupled inbetween the receiver (531) and an entropy decoder/parser (520) (“parser(520)” henceforth). In certain applications, the buffer memory (515) ispart of the video decoder (510). In others, it can be outside of thevideo decoder (510) (not depicted). In still others, there can be abuffer memory (not depicted) outside of the video decoder (510), forexample to combat network jitter, and in addition another buffer memory(515) inside the video decoder (510), for example to handle playouttiming. When the receiver (531) is receiving data from a store/forwarddevice of sufficient bandwidth and controllability, or from anisosynchronous network, the buffer memory (515) may not be needed, orcan be small. For use on best effort packet networks such as theInternet, the buffer memory (515) may be required, can be comparativelylarge and can be advantageously of adaptive size, and may at leastpartially be implemented in an operating system or similar elements (notdepicted) outside of the video decoder (510).

The video decoder (510) may include the parser (520) to reconstructsymbols (521) from the coded video sequence. Categories of those symbolsinclude information used to manage operation of the video decoder (510),and potentially information to control a rendering device such as arender device (512) (e.g., a display screen) that is not an integralpart of the electronic device (530) but can be coupled to the electronicdevice (530), as was shown in FIG. 5. The control information for therendering device(s) may be in the form of Supplemental EnhancementInformation (SEI messages) or Video Usability Information (VUI)parameter set fragments (not depicted). The parser (520) mayparse/entropy-decode the coded video sequence that is received. Thecoding of the coded video sequence can be in accordance with a videocoding technology or standard, and can follow various principles,including variable length coding, Huffman coding, arithmetic coding withor without context sensitivity, and so forth. The parser (520) mayextract from the coded video sequence, a set of subgroup parameters forat least one of the subgroups of pixels in the video decoder, based uponat least one parameter corresponding to the group. Subgroups can includeGroups of Pictures (GOPs), pictures, tiles, slices, macroblocks, CodingUnits (CUs), blocks, Transform Units (TUs), Prediction Units (PUs) andso forth. The parser (520) may also extract from the coded videosequence information such as transform coefficients, quantizer parametervalues, motion vectors, and so forth.

The parser (520) may perform an entropy decoding/parsing operation onthe video sequence received from the buffer memory (515), so as tocreate symbols (521).

Reconstruction of the symbols (521) can involve multiple different unitsdepending on the type of the coded video picture or parts thereof (suchas: inter and intra picture, inter and intra block), and other factors.Which units are involved, and how, can be controlled by the subgroupcontrol information that was parsed from the coded video sequence by theparser (520). The flow of such subgroup control information between theparser (520) and the multiple units below is not depicted for clarity.

Beyond the functional blocks already mentioned, the video decoder (510)can be conceptually subdivided into a number of functional units asdescribed below. In a practical implementation operating undercommercial constraints, many of these units interact closely with eachother and can, at least partly, be integrated into each other. However,for the purpose of describing the disclosed subject matter, theconceptual subdivision into the functional units below is appropriate.

A first unit is the scaler/inverse transform unit (551). Thescaler/inverse transform unit (551) receives a quantized transformcoefficient as well as control information, including which transform touse, block size, quantization factor, quantization scaling matrices,etc. as symbol(s) (521) from the parser (520). The scaler/inversetransform unit (551) can output blocks comprising sample values, thatcan be input into aggregator (555).

In some cases, the output samples of the scaler/inverse transform (551)can pertain to an intra coded block; that is: a block that is not usingpredictive information from previously reconstructed pictures, but canuse predictive information from previously reconstructed parts of thecurrent picture. Such predictive information can be provided by an intrapicture prediction unit (552). In some cases, the intra pictureprediction unit (552) generates a block of the same size and shape ofthe block under reconstruction, using surrounding already reconstructedinformation fetched from the current picture buffer (558). The currentpicture buffer (558) buffers, for example, partly reconstructed currentpicture and/or fully reconstructed current picture. The aggregator(555), in some cases, adds, on a per sample basis, the predictioninformation the intra prediction unit (552) has generated to the outputsample information as provided by the scaler/inverse transform unit(551).

In other cases, the output samples of the scaler/inverse transform unit(551) can pertain to an inter coded, and potentially motion compensatedblock. In such a case, a motion compensation prediction unit (553) canaccess reference picture memory (557) to fetch samples used forprediction. After motion compensating the fetched samples in accordancewith the symbols (521) pertaining to the block, these samples can beadded by the aggregator (555) to the output of the scaler/inversetransform unit (551) (in this case called the residual samples orresidual signal) so as to generate output sample information. Theaddresses within the reference picture memory (557) from where themotion compensation prediction unit (553) fetches prediction samples canbe controlled by motion vectors, available to the motion compensationprediction unit (553) in the form of symbols (521) that can have, forexample X, Y, and reference picture components. Motion compensation alsocan include interpolation of sample values as fetched from the referencepicture memory (557) when sub-sample exact motion vectors are in use,motion vector prediction mechanisms, and so forth.

The output samples of the aggregator (555) can be subject to variousloop filtering techniques in the loop filter unit (556). Videocompression technologies can include in-loop filter technologies thatare controlled by parameters included in the coded video sequence (alsoreferred to as coded video bitstream) and made available to the loopfilter unit (556) as symbols (521) from the parser (520), but can alsobe responsive to meta-information obtained during the decoding ofprevious (in decoding order) parts of the coded picture or coded videosequence, as well as responsive to previously reconstructed andloop-filtered sample values.

The output of the loop filter unit (556) can be a sample stream that canbe output to the render device (512) as well as stored in the referencepicture memory (557) for use in future inter-picture prediction.

Certain coded pictures, once fully reconstructed, can be used asreference pictures for future prediction. For example, once a codedpicture corresponding to a current picture is fully reconstructed andthe coded picture has been identified as a reference picture (by, forexample, the parser (520)), the current picture buffer (558) can becomea part of the reference picture memory (557), and a fresh currentpicture buffer can be reallocated before commencing the reconstructionof the following coded picture.

The video decoder (510) may perform decoding operations according to apredetermined video compression technology in a standard, such as ITU-TRec. H.265. The coded video sequence may conform to a syntax specifiedby the video compression technology or standard being used, in the sensethat the coded video sequence adheres to both the syntax of the videocompression technology or standard and the profiles as documented in thevideo compression technology or standard. Specifically, a profile canselect certain tools as the only tools available for use under thatprofile from all the tools available in the video compression technologyor standard. Also necessary for compliance can be that the complexity ofthe coded video sequence is within bounds as defined by the level of thevideo compression technology or standard. In some cases, levels restrictthe maximum picture size, maximum frame rate, maximum reconstructionsample rate (measured in, for example megasamples per second), maximumreference picture size, and so on. Limits set by levels can, in somecases, be further restricted through Hypothetical Reference Decoder(HRD) specifications and metadata for HRD buffer management signaled inthe coded video sequence.

In an embodiment, the receiver (531) may receive additional (redundant)data with the encoded video. The additional data may be included as partof the coded video sequence(s). The additional data may be used by thevideo decoder (510) to properly decode the data and/or to moreaccurately reconstruct the original video data. Additional data can bein the form of, for example, temporal, spatial, or signal noise ratio(SNR) enhancement layers, redundant slices, redundant pictures, forwarderror correction codes, and so on.

FIG. 6 shows a block diagram of a video encoder (603) according to anembodiment of the present disclosure. The video encoder (603) isincluded in an electronic device (620). The electronic device (620)includes a transmitter (640) (e.g., transmitting circuitry). The videoencoder (603) can be used in the place of the video encoder (403) in theFIG. 4 example.

The video encoder (603) may receive video samples from a video source(601) (that is not part of the electronic device (620) in the FIG. 6example) that may capture video image(s) to be coded by the videoencoder (603). In another example, the video source (601) is a part ofthe electronic device (620).

The video source (601) may provide the source video sequence to be codedby the video encoder (603) in the form of a digital video sample streamthat can be of any suitable bit depth (for example: 8 bit, 10 bit, 12bit, . . . ), any colorspace (for example, BT.601 Y CrCB, RGB, . . . ),and any suitable sampling structure (for example Y CrCb 4:2:0, Y CrCb4:4:4). In a media serving system, the video source (601) may be astorage device storing previously prepared video. In a videoconferencingsystem, the video source (601) may be a camera that captures local imageinformation as a video sequence. Video data may be provided as aplurality of individual pictures that impart motion when viewed insequence. The pictures themselves may be organized as a spatial array ofpixels, wherein each pixel can comprise one or more samples depending onthe sampling structure, color space, etc. in use. A person skilled inthe art can readily understand the relationship between pixels andsamples. The description below focuses on samples.

According to an embodiment, the video encoder (603) may code andcompress the pictures of the source video sequence into a coded videosequence (643) in real time or under any other time constraints asrequired by the application. Enforcing appropriate coding speed is onefunction of a controller (650). In some embodiments, the controller(650) controls other functional units as described below and isfunctionally coupled to the other functional units. The coupling is notdepicted for clarity. Parameters set by the controller (650) can includerate control related parameters (picture skip, quantizer, lambda valueof rate-distortion optimization techniques, . . . ), picture size, groupof pictures (GOP) layout, maximum motion vector search range, and soforth. The controller (650) can be configured to have other suitablefunctions that pertain to the video encoder (603) optimized for acertain system design.

In some embodiments, the video encoder (603) is configured to operate ina coding loop. As an oversimplified description, in an example, thecoding loop can include a source coder (630) (e.g., responsible forcreating symbols, such as a symbol stream, based on an input picture tobe coded, and a reference picture(s)), and a (local) decoder (633)embedded in the video encoder (603). The decoder (633) reconstructs thesymbols to create the sample data in a similar manner as a (remote)decoder also would create (as any compression between symbols and codedvideo bitstream is lossless in the video compression technologiesconsidered in the disclosed subject matter). The reconstructed samplestream (sample data) is input to the reference picture memory (634). Asthe decoding of a symbol stream leads to bit-exact results independentof decoder location (local or remote), the content in the referencepicture memory (634) is also bit exact between the local encoder andremote encoder. In other words, the prediction part of an encoder “sees”as reference picture samples exactly the same sample values as a decoderwould “see” when using prediction during decoding. This fundamentalprinciple of reference picture synchronicity (and resulting drift, ifsynchronicity cannot be maintained, for example because of channelerrors) is used in some related arts as well.

The operation of the “local” decoder (633) can be the same as of a“remote” decoder, such as the video decoder (510), which has alreadybeen described in detail above in conjunction with FIG. 5. Brieflyreferring also to FIG. 5, however, as symbols are available andencoding/decoding of symbols to a coded video sequence by an entropycoder (645) and the parser (520) can be lossless, the entropy decodingparts of the video decoder (510), including the buffer memory (515), andparser (520) may not be fully implemented in the local decoder (633).

An observation that can be made at this point is that any decodertechnology except the parsing/entropy decoding that is present in adecoder also necessarily needs to be present, in substantially identicalfunctional form, in a corresponding encoder. For this reason, thedisclosed subject matter focuses on decoder operation. The descriptionof encoder technologies can be abbreviated as they are the inverse ofthe comprehensively described decoder technologies. Only in certainareas a more detail description is required and provided below.

During operation, in some examples, the source coder (630) may performmotion compensated predictive coding, which codes an input picturepredictively with reference to one or more previously-coded picture fromthe video sequence that were designated as “reference pictures”. In thismanner, the coding engine (632) codes differences between pixel blocksof an input picture and pixel blocks of reference picture(s) that may beselected as prediction reference(s) to the input picture.

The local video decoder (633) may decode coded video data of picturesthat may be designated as reference pictures, based on symbols createdby the source coder (630). Operations of the coding engine (632) mayadvantageously be lossy processes. When the coded video data may bedecoded at a video decoder (not shown in FIG. 6), the reconstructedvideo sequence typically may be a replica of the source video sequencewith some errors. The local video decoder (633) replicates decodingprocesses that may be performed by the video decoder on referencepictures and may cause reconstructed reference pictures to be stored inthe reference picture cache (634). In this manner, the video encoder(603) may store copies of reconstructed reference pictures locally thathave common content as the reconstructed reference pictures that will beobtained by a far-end video decoder (absent transmission errors).

The predictor (635) may perform prediction searches for the codingengine (632). That is, for a new picture to be coded, the predictor(635) may search the reference picture memory (634) for sample data (ascandidate reference pixel blocks) or certain metadata such as referencepicture motion vectors, block shapes, and so on, that may serve as anappropriate prediction reference for the new pictures. The predictor(635) may operate on a sample block-by-pixel block basis to findappropriate prediction references. In some cases, as determined bysearch results obtained by the predictor (635), an input picture mayhave prediction references drawn from multiple reference pictures storedin the reference picture memory (634).

The controller (650) may manage coding operations of the source coder(630), including, for example, setting of parameters and subgroupparameters used for encoding the video data.

Output of all aforementioned functional units may be subjected toentropy coding in the entropy coder (645). The entropy coder (645)translates the symbols as generated by the various functional units intoa coded video sequence, by lossless compressing the symbols according totechnologies such as Huffman coding, variable length coding, arithmeticcoding, and so forth.

The transmitter (640) may buffer the coded video sequence(s) as createdby the entropy coder (645) to prepare for transmission via acommunication channel (660), which may be a hardware/software link to astorage device which would store the encoded video data. The transmitter(640) may merge coded video data from the video coder (603) with otherdata to be transmitted, for example, coded audio data and/or ancillarydata streams (sources not shown).

The controller (650) may manage operation of the video encoder (603).During coding, the controller (650) may assign to each coded picture acertain coded picture type, which may affect the coding techniques thatmay be applied to the respective picture. For example, pictures oftenmay be assigned as one of the following picture types:

An Intra Picture (I picture) may be one that may be coded and decodedwithout using any other picture in the sequence as a source ofprediction. Some video codecs allow for different types of intrapictures, including, for example Independent Decoder Refresh (“IDR”)Pictures. A person skilled in the art is aware of those variants of Ipictures and their respective applications and features.

A predictive picture (P picture) may be one that may be coded anddecoded using intra prediction or inter prediction using at most onemotion vector and reference index to predict the sample values of eachblock.

A bi-directionally predictive picture (B Picture) may be one that may becoded and decoded using intra prediction or inter prediction using atmost two motion vectors and reference indices to predict the samplevalues of each block. Similarly, multiple-predictive pictures can usemore than two reference pictures and associated metadata for thereconstruction of a single block.

Source pictures commonly may be subdivided spatially into a plurality ofsample blocks (for example, blocks of 4×4, 8×8, 4×8, or 16×16 sampleseach) and coded on a block-by-block basis. Blocks may be codedpredictively with reference to other (already coded) blocks asdetermined by the coding assignment applied to the blocks' respectivepictures. For example, blocks of I pictures may be codednon-predictively or they may be coded predictively with reference toalready coded blocks of the same picture (spatial prediction or intraprediction). Pixel blocks of P pictures may be coded predictively, viaspatial prediction or via temporal prediction with reference to onepreviously coded reference picture. Blocks of B pictures may be codedpredictively, via spatial prediction or via temporal prediction withreference to one or two previously coded reference pictures.

The video encoder (603) may perform coding operations according to apredetermined video coding technology or standard, such as ITU-T Rec.H.265. In its operation, the video encoder (603) may perform variouscompression operations, including predictive coding operations thatexploit temporal and spatial redundancies in the input video sequence.The coded video data, therefore, may conform to a syntax specified bythe video coding technology or standard being used.

In an embodiment, the transmitter (640) may transmit additional datawith the encoded video. The source coder (630) may include such data aspart of the coded video sequence. Additional data may comprisetemporal/spatial/SNR enhancement layers, other forms of redundant datasuch as redundant pictures and slices, SEI messages, VUI parameter setfragments, and so on.

A video may be captured as a plurality of source pictures (videopictures) in a temporal sequence. Intra-picture prediction (oftenabbreviated to intra prediction) makes use of spatial correlation in agiven picture, and inter-picture prediction makes uses of the (temporalor other) correlation between the pictures. In an example, a specificpicture under encoding/decoding, which is referred to as a currentpicture, is partitioned into blocks. When a block in the current pictureis similar to a reference block in a previously coded and still bufferedreference picture in the video, the block in the current picture can becoded by a vector that is referred to as a motion vector. The motionvector points to the reference block in the reference picture, and canhave a third dimension identifying the reference picture, in casemultiple reference pictures are in use.

In some embodiments, a bi-prediction technique can be used in theinter-picture prediction. According to the bi-prediction technique, tworeference pictures, such as a first reference picture and a secondreference picture that are both prior in decoding order to the currentpicture in the video (but may be in the past and future, respectively,in display order) are used. A block in the current picture can be codedby a first motion vector that points to a first reference block in thefirst reference picture, and a second motion vector that points to asecond reference block in the second reference picture. The block can bepredicted by a combination of the first reference block and the secondreference block.

Further, a merge mode technique can be used in the inter-pictureprediction to improve coding efficiency.

According to some embodiments of the disclosure, predictions, such asinter-picture predictions and intra-picture predictions are performed inthe unit of blocks. For example, according to the HEVC standard, apicture in a sequence of video pictures is partitioned into coding treeunits (CTU) for compression, the CTUs in a picture have the same size,such as 64×64 pixels, 32×32 pixels, or 16×16 pixels. In general, a CTUincludes three coding tree blocks (CTBs), which are one luma CTB and twochroma CTBs. Each CTU can be recursively quadtree split into one ormultiple coding units (CUs). For example, a CTU of 64×64 pixels can besplit into one CU of 64×64 pixels, or 4 CUs of 32×32 pixels, or 16 CUsof 16×16 pixels. In an example, each CU is analyzed to determine aprediction type for the CU, such as an inter prediction type or an intraprediction type. The CU is split into one or more prediction units (PUs)depending on the temporal and/or spatial predictability. Generally, eachPU includes a luma prediction block (PB), and two chroma PBs. In anembodiment, a prediction operation in coding (encoding/decoding) isperformed in the unit of a prediction block. Using a luma predictionblock as an example of a prediction block, the prediction block includesa matrix of values (e.g., luma values) for pixels, such as 8×8 pixels,16×16 pixels, 8×16 pixels, 16×8 pixels, and the like.

FIG. 7 shows a diagram of a video encoder (703) according to anotherembodiment of the disclosure. The video encoder (703) is configured toreceive a processing block (e.g., a prediction block) of sample valueswithin a current video picture in a sequence of video pictures, andencode the processing block into a coded picture that is part of a codedvideo sequence. In an example, the video encoder (703) is used in theplace of the video encoder (403) in the FIG. 4 example.

In an HEVC example, the video encoder (703) receives a matrix of samplevalues for a processing block, such as a prediction block of 8×8samples, and the like. The video encoder (703) determines whether theprocessing block is best coded using intra mode, inter mode, orbi-prediction mode using, for example, rate-distortion optimization.When the processing block is to be coded in intra mode, the videoencoder (703) may use an intra prediction technique to encode theprocessing block into the coded picture; and when the processing blockis to be coded in inter mode or bi-prediction mode, the video encoder(703) may use an inter prediction or bi-prediction technique,respectively, to encode the processing block into the coded picture. Incertain video coding technologies, merge mode can be an inter pictureprediction submode where the motion vector is derived from one or moremotion vector predictors without the benefit of a coded motion vectorcomponent outside the predictors. In certain other video codingtechnologies, a motion vector component applicable to the subject blockmay be present. In an example, the video encoder (703) includes othercomponents, such as a mode decision module (not shown) to determine themode of the processing blocks.

In the FIG. 7 example, the video encoder (703) includes the interencoder (730), an intra encoder (722), a residue calculator (723), aswitch (726), a residue encoder (724), a general controller (721), andan entropy encoder (725) coupled together as shown in FIG. 7.

The inter encoder (730) is configured to receive the samples of thecurrent block (e.g., a processing block), compare the block to one ormore reference blocks in reference pictures (e.g., blocks in previouspictures and later pictures), generate inter prediction information(e.g., description of redundant information according to inter encodingtechnique, motion vectors, merge mode information), and calculate interprediction results (e.g., predicted block) based on the inter predictioninformation using any suitable technique. In some examples, thereference pictures are decoded reference pictures that are decoded basedon the encoded video information.

The intra encoder (722) is configured to receive the samples of thecurrent block (e.g., a processing block), in some cases compare theblock to blocks already coded in the same picture, generate quantizedcoefficients after transform, and in some cases also intra predictioninformation (e.g., an intra prediction direction information accordingto one or more intra encoding techniques). In an example, the intraencoder (722) also calculates intra prediction results (e.g., predictedblock) based on the intra prediction information and reference blocks inthe same picture.

The general controller (721) is configured to determine general controldata and control other components of the video encoder (703) based onthe general control data. In an example, the general controller (721)determines the mode of the block, and provides a control signal to theswitch (726) based on the mode. For example, when the mode is the intramode, the general controller (721) controls the switch (726) to selectthe intra mode result for use by the residue calculator (723), andcontrols the entropy encoder (725) to select the intra predictioninformation and include the intra prediction information in thebitstream; and when the mode is the inter mode, the general controller(721) controls the switch (726) to select the inter prediction resultfor use by the residue calculator (723), and controls the entropyencoder (725) to select the inter prediction information and include theinter prediction information in the bitstream.

The residue calculator (723) is configured to calculate a difference(residue data) between the received block and prediction resultsselected from the intra encoder (722) or the inter encoder (730). Theresidue encoder (724) is configured to operate based on the residue datato encode the residue data to generate the transform coefficients. In anexample, the residue encoder (724) is configured to convert the residuedata from a spatial domain to a frequency domain, and generate thetransform coefficients. The transform coefficients are then subject toquantization processing to obtain quantized transform coefficients. Invarious embodiments, the video encoder (703) also includes a residuedecoder (728). The residue decoder (728) is configured to performinverse-transform, and generate the decoded residue data. The decodedresidue data can be suitably used by the intra encoder (722) and theinter encoder (730). For example, the inter encoder (730) can generatedecoded blocks based on the decoded residue data and inter predictioninformation, and the intra encoder (722) can generate decoded blocksbased on the decoded residue data and the intra prediction information.The decoded blocks are suitably processed to generate decoded picturesand the decoded pictures can be buffered in a memory circuit (not shown)and used as reference pictures in some examples.

The entropy encoder (725) is configured to format the bitstream toinclude the encoded block. The entropy encoder (725) is configured toinclude various information according to a suitable standard, such asthe HEVC standard. In an example, the entropy encoder (725) isconfigured to include the general control data, the selected predictioninformation (e.g., intra prediction information or inter predictioninformation), the residue information, and other suitable information inthe bitstream. Note that, according to the disclosed subject matter,when coding a block in the merge submode of either inter mode orbi-prediction mode, there is no residue information.

FIG. 8 shows a diagram of a video decoder (810) according to anotherembodiment of the disclosure. The video decoder (810) is configured toreceive coded pictures that are part of a coded video sequence, anddecode the coded pictures to generate reconstructed pictures. In anexample, the video decoder (810) is used in the place of the videodecoder (410) in the FIG. 4 example.

In the FIG. 8 example, the video decoder (810) includes an entropydecoder (871), an inter decoder (880), a residue decoder (873), areconstruction module (874), and an intra decoder (872) coupled togetheras shown in FIG. 8.

The entropy decoder (871) can be configured to reconstruct, from thecoded picture, certain symbols that represent the syntax elements ofwhich the coded picture is made up. Such symbols can include, forexample, the mode in which a block is coded (such as, for example, intramode, inter mode, bi-predicted mode, the latter two in merge submode oranother submode), prediction information (such as, for example, intraprediction information or inter prediction information) that canidentify certain sample or metadata that is used for prediction by theintra decoder (872) or the inter decoder (880), respectively, residualinformation in the form of, for example, quantized transformcoefficients, and the like. In an example, when the prediction mode isinter or bi-predicted mode, the inter prediction information is providedto the inter decoder (880); and when the prediction type is the intraprediction type, the intra prediction information is provided to theintra decoder (872). The residual information can be subject to inversequantization and is provided to the residue decoder (873).

The inter decoder (880) is configured to receive the inter predictioninformation, and generate inter prediction results based on the interprediction information.

The intra decoder (872) is configured to receive the intra predictioninformation, and generate prediction results based on the intraprediction information.

The residue decoder (873) is configured to perform inverse quantizationto extract de-quantized transform coefficients, and process thede-quantized transform coefficients to convert the residual from thefrequency domain to the spatial domain. The residue decoder (873) mayalso require certain control information (to include the QuantizerParameter (QP)), and that information may be provided by the entropydecoder (871) (data path not depicted as this may be low volume controlinformation only).

The reconstruction module (874) is configured to combine, in the spatialdomain, the residual as output by the residue decoder (873) and theprediction results (as output by the inter or intra prediction modulesas the case may be) to form a reconstructed block, that may be part ofthe reconstructed picture, which in turn may be part of thereconstructed video. It is noted that other suitable operations, such asa deblocking operation and the like, can be performed to improve thevisual quality.

It is noted that the video encoders (403), (603), and (703), and thevideo decoders (410), (510), and (810) can be implemented using anysuitable technique. In an embodiment, the video encoders (403), (603),and (703), and the video decoders (410), (510), and (810) can beimplemented using one or more integrated circuits. In anotherembodiment, the video encoders (403), (603), and (603), and the videodecoders (410), (510), and (810) can be implemented using one or moreprocessors that execute software instructions.

In some embodiments, such as in HEVC, a primary transform may include4-point, 8-point, 16-point and 32-point discrete cosine transform (DCT)type 2 (DCT-2), and the transform core matrices may be represented using8-bit integers (i.e., 8-bit transform core). The transform core matricesof a smaller DCT-2 are part of transform core matrices of a largerDCT-2, as shown in APPENDIX I.

The DCT-2 core matrices show symmetry/anti-symmetry characteristics.Therefore, a “partial butterfly” implementation may be supported toreduce a number of operation counts (e.g., multiplications, additions,subtractions, shifts, and/or the like), and identical results of matrixmultiplication can be obtained using the partial butterfly.

In some embodiments, such as in VVC, besides the 4-point, 8-point,16-point, and 32-point DCT-2 transforms described above, additional2-point and 64-point DCT-2 may also be included. An example of a64-point DCT-2 core, such as used in VVC, is shown in APPENDIX II as a64×64 matrix.

In addition to DCT-2 and 4×4 DST-7, such as used in HEVC, an AdaptiveMultiple Transform (AMT) (also known as Enhanced Multiple Transform(EMT) or Multiple Transform Selection (MTS)) scheme, can be used, suchas in VVC, for residual coding for both inter and intra coded blocks.The AMT scheme may use multiple selected transforms from the DCT/DSTfamilies other than the current transforms in HEVC. The newly introducedtransform matrices are DST-7, DCT-8. Table 1 shows examples of the basisfunctions of the selected DST/DCT for an N-point input.

TABLE 1 Transform Type Basis function T_(i)(j), i, j = 0, 1, . . . , N−1DCT-2${T_{i}(j)} = {\omega_{0} \cdot \sqrt{\frac{2}{N}} \cdot {\cos\left( \frac{\pi \cdot i \cdot \left( {{2\; j} + 1} \right)}{2\; N} \right)}}$${{where}\mspace{14mu}\omega_{0}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix} \right.$ DCT-8${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\cos\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {{2\; j} + 1} \right)}{{4\; N} + 2} \right)}}$DST-7${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {j + 1} \right)}{{2N} + 1} \right)}}$

The primary transform matrices, such as used in VVC, may be used with a8-bit representation. In an embodiment, the AMT applies transformmatrices to the CUs with both a width and a height smaller than or equalto 32. Whether AMT is applied may be controlled by a flag (e.g., anmts_flag). When the mts_flag is equal to 0, in some examples, only DCT-2is applied for coding residue data. When the mts_flag is equal to 1, anindex (e.g., an mts_idx) may be further signalled using 2 bins toidentify the horizontal and vertical transform to be used according toTable 2 for example, where a type value of 1 means DST-7 is used, andtype value of 2 means DCT-8 is used. In Table 2, the specification oftrTypeHor and trTypeVer depend on mts_idx[x][y][cIdx].

TABLE 2 mts_idx[ xTbY ][ yTbY ][ cIdx ] trTypeHor trTypeVer −1 0 0 0 1 11 2 1 2 1 2 3 2 2

In some embodiments, an implicit MTS can be applied when the abovesignaling based MTS (i.e., explicit MTS) is not used. With the implicitMTS, the transform selection is made according to the block width andheight instead of the signaling. For example, with an implicit MTS,DST-7 is selected for a shorter side (i.e., a minimum one of M and N) ofthe block of M×N and DCT-2 is selected for a longer side (i.e., amaximum one of M and N) of the block.

Exemplary transform cores, each of which is a matrix composed by thebasis vectors, of DST-7 and DCT-8 are illustrated in APPENDIX III.

In some examples, such as in VVC, when both the height and width of thecoding block is smaller than or equal to 64, the TB size is the same asthe coding block size. When either the height or width of the codingblock is larger than 64, when doing a transform (such as an inversetransform, an inverse primary transform, or the like) or intraprediction, the coding block is further split into multiple sub-blocks,where the width and height of each sub-block is smaller than or equal to64. One transform can be performed on each sub-block.

Related syntax and semantics of MTS in some examples in VVC can bedescribed below (highlighted using a label 910 and a label 1010) inFIGS. 9 and 10A-10C. An example of transform unit syntax is shown inFIG. 9. An example of a residual coding syntax is shown in FIGS.10A-10C.

An example of the transform unit semantics is as follows.cu_mts_flag[x0][y0] equal to 1 specifies that multiple transformselection is applied to the residual samples of the associated lumatransform block. cu_mts_flag[x0][y0] equal to 0 specifies that multipletransform selection is not applied to the residual samples of theassociated luma transform block. The array indices x0, y0 specify thelocation (x0, y0) of the top-left luma sample of the consideredtransform block relative to the top-left luma sample of the picture.When cu_mts_flag[x0][y0] is not present, it is inferred to be equal to0.

An example of the residual coding semantics is as follows.mts_idx[x0][y0] specifies which transform kernels are applied to theluma residual samples along the horizontal and vertical direction of thecurrent transform block. The array indices x0, y0 specify the location(x0, y0) of the top-left luma sample of the considered transform blockrelative to the top-left luma sample of the picture. Whenmts_idx[x0][y0] is not present, it is inferred to be equal to −1.

FIG. 11A shows an exemplary forward transform (also referred to as aforward primary transform) performed by an encoder. The forwardtransform can include a forward horizontal transform and a forwardvertical transform. The forward horizontal transform is applied first toa residual block (1110) having residual data to obtain an intermediateblock. Subsequently, the forward vertical transform is applied to theintermediate block to obtain a coefficient block (1112) having transformcoefficients.

FIG. 11B shows an exemplary backward transform (also referred to as aninverse primary transform or an inverse transform) performed by adecoder. Generally speaking, the inverse transform matches the forwardtransform. The inverse primary transform can include an inverse primaryhorizontal transform (also referred to as an inverse horizontaltransform) and an inverse primary vertical transform (also referred toas an inverse vertical transform). To match the forward transform, anorder of applying the inverse horizontal and vertical transforms isswitched in the inverse transform. Accordingly, the inverse verticaltransform is applied first to a coefficient block (1122) to obtain anintermediate block. Subsequently, the inverse horizontal transform isapplied to the intermediate block to obtain a residual block (1120).

A primary transform can refer to a forward primary transform or aninverse primary transform. A horizontal transform can refer to aninverse horizontal transform or a forward horizontal transform.Similarly, a vertical transform can refer to an inverse verticaltransform or a forward vertical transform.

In an example, such as in VVC, at the decoder, the inverse verticalprimary transform (also referred to as inverse vertical transform) isperformed first, then the inverse horizontal primary transform (alsoreferred to as inverse horizontal transform) is performed second afterapplying the inverse vertical transform, as shown in FIG. 11B and intexts highlighted using labels 1210-1211 in FIGS. 12A-12E. An example ofa transformation process for scaled transform coefficients is shown inFIGS. 12A-12E.

A primary transform, such as a forward primary transform or an inverseprimary transform can utilize a zero-out method or zero-out scheme asdescribed below. In some examples, such as in VVC, for a 64-point (or64-length) DCT-2, only the first 32 coefficients are calculated and theremaining coefficients are set as 0. Therefore, for an M×N block whichis coded using a DCT-2 transform, the top-left min(M, 32)×min(N,32) lowfrequency coefficients are calculated. The remaining coefficients areset as 0 and not signaled. In an example, the remaining coefficients arenot calculated. The entropy coding of the coefficient block can beperformed by setting the coefficient block size as min(M, 32)×min(N,32),such that the coefficient coding of an M×N block is regarded as a min(M,32)×min(N, 32) coefficient block.

In some examples in which MTS is used, for 32-point DST-7 or DCT-8, onlythe first 16 coefficients are calculated and the remaining coefficientsare set as 0. Therefore, for an M×N block which is coded using a DST-7or DCT-8 transform, the top-left min(M, 16)×min(N, 16) low frequencycoefficients are kept. The remaining coefficients can be set as 0 andnot signaled. However, different from the coefficient coding schemewhich is used when 64-point zero-out DCT-2 is applied, for 32-point MTS,the coefficient coding is still performed on the whole M×N block evenwhen M or N is larger than 16. However, when a coefficient group (CG) isoutside the top-left 16×16 low-frequency region (i.e., the coefficientgroup is in a zero-out region) a flag indicating whether the coefficientgroup has a nonzero coefficient (e.g., a coded_sub_block_flag) is notsignaled. The zero-out region refers to a region in the coefficientblock where coefficients are zero, and thus the coefficients in thezero-out region are zero. An example of a residual coding syntax isdescribed below, indicated by the highlighted texts using label 1310 inFIG. 13.

One major complexity aspect of a primary transform, such as an inverseprimary transform or a forward primary transform, is the average numberof multiplications per sample (MPS). For example, for an 8×8 primarytransform, if DST-7 using a full matrix multiplication is applied forboth the horizontal and vertical transform for the 8×8 block, thehorizontal transform requires 8 multiplications to calculate eachcoefficient and the vertical transform requires 8 multiplications tocalculate each coefficient, so together 16 multiplications per sample isused.

In some examples, a worst-case MPS is 32×32 DCT2 and 32×64 DCT-2, asdescribed below. A MPS can also be referred to as multiplications percoefficient.

For a M×N TU, if non-zero transform coefficients are kept for thetop-left mxn (i.e., m by n) region of the TU, the number ofmultiplications per coefficient for the inverse transform implemented inthe direct matrix multiply structure can be computed by

${{Muls}_{-}{per}_{-}{coeff}_{directMatrixMultiply}} = {{\frac{m}{M}n} + {m.}}$When n is less than N or m is less than M, the zero-out method isapplied and the number of multiplications per coefficient can bereduced.

Likewise, the number of multiplications per coefficient for the inversetransform implemented in the structure of half butterfly and half directmatrix multiply can be computed by

${{Muls}_{-}{per}_{-}{coeff}_{halfButterfly}} = {{\frac{m}{M}\left( {\frac{n}{2} + {\log_{2}\frac{n}{2}}} \right)} + {\left( {\frac{m}{2} + {\log_{2}\frac{m}{2}}} \right).}}$

In some examples, such as in VVC, DST7/DCT8 transforms up to 32-point(e.g., 4-point DST-7, 8-point DST-7, 16-point DST-7, 32-point DST-7,4-point DCT-8, 8-point DCT-8, 16-point DCT-8, and 32-point DCT-8), andDCT2 transforms up to 64-point are defined for primary transforms (e.g.,4-point DCT-2, 8-point DCT-2, 16-point DCT-2, 32-point DCT-2, and64-point DCT-2). For the 64-point primary transform (e.g., 64-pointDCT-2), a transform coefficient zero-out scheme (also referred tozero-out method, zero-out scheme) can be applied. With the zero-outscheme, for a 64-point primary transform, in some examples, only thefirst 32 coefficient are kept, and the remaining coefficients are set as0. For example, for a 64×64 block, the top-left 32×32 coefficients arekept, and the remaining coefficients are set as 0. For a 64×N block,where N is less than or equal to 32, the left 32×N coefficients may bekept, and the remaining coefficients may be set as 0. For a M×64 block,where M is less than or equal to 32, the top M×32 coefficients may bekept and the remaining coefficients are set as 0.

Table 3 shows numbers of multiplications per coefficient where M is theblock width, N is the block height, m is the number of coefficients keptin the horizontal direction (i.e., in each row), and n is the number ofcoefficients kept in the vertical direction (i.e., in each column). Asillustrated in Table 3, when the top-left 16×16 coefficients are keptfor DST-7/DCT-8 (i.e., m=n=16), the worst case (e.g., the number ofmultiplications per coefficient is a maximum number) includes 32×64 and32×32 DCT-2. For example, the number of multiplications per coefficientfor 32×64 and 32×32 DCT-2 is 40 (using half butterfly), and correspondsto the worse case.

In some examples, 32×32 DCT2 without zero-out and using half butterfly,32×32 DST7/DCT8 with zero-out (m=n=16), 32×16 DST7/DCT8 with zero-out(m=n=16), 16×32 DST7/DCT8 with zero-out (m=n=16) can be employed invideo coding.

TABLE 3 number of multiplications number of per coefficientmultiplications (direct matrix per coefficient M N m n multiply) (halfbutterfly) DCT2 TU size M*N, and keep the top-left m*n non-zerocoefficients 64 64 32 32 48 30 64 32 32 32 48 30 32 64 32 32 64 40 32 3232 32 64 40 DST7/DCT8 TU size M*N, and keep the top-left m*n non-zerocoefficients 32 32 32 32 64 32 16 32 16 48 16 32 16 32 48 32 32 16 16 2432 16 16 16 24 16 32 16 16 32

In an embodiment, a mode-dependent non-separable secondary transform(NSST) can be used between a forward core transform and a quantizationat an encoder side and between a de-quantization and an inverse coretransform at a decoder side. For example, to keep a low complexity, aNSST is applied to low frequency coefficients after a primary transform(or a core transform). When both a width (W) and a height (H) of atransform coefficient block are larger than or equal to 8, an 8×8 NSSTis applied to a top-left 8×8 region of the transform coefficients block.Otherwise, when either the width W or the height H of the transformcoefficient block is 4, a 4×4 NSST is applied, and the 4×4 NSST isperformed on a top-left min(8,W)×min(8,H) region of the transformcoefficient block. The above transform selection method is applied forboth luma and chroma components.

A matrix multiplication implementation of a NSST is described as followsusing a 4×4 input block as an example. The 4×4 input block X is writtenin Eq. (1) as

$\begin{matrix}{X = \begin{bmatrix}X_{00} & X_{01} & X_{02} & X_{03} \\X_{10} & X_{11} & X_{12} & X_{13} \\X_{20} & X_{21} & X_{22} & X_{23} \\X_{30} & X_{31} & X_{32} & X_{33}\end{bmatrix}} & (1)\end{matrix}$

The input block X can be represented as a vector

in Eq. (2) where{right arrow over (X)}=[X ₀₀ X ₀₁ X ₀₂ X ₀₃ X ₁₀ X ₁₁ X ₁₂ X ₁₃ X ₂₀ X₂₁ X ₂₂ X ₂₃ X ₃₀ X ₃₁ X ₃₂ X ₃₃]^(T)  (2)

The non-separable transform is calculated as

=T·

, where

indicates a transform coefficient vector, and T is a 16×16 transformmatrix. The 16×1 transform coefficient vector

is subsequently reorganized as a 4×4 block using a scanning order (forexample, a horizontal scanning order, a vertical scanning order or adiagonal scanning order) for the input block X. Coefficients withsmaller indices can be placed with smaller scanning indices in the 4×4coefficient block. In some embodiments, a Hypercube-Givens Transform(HyGT) with a butterfly implementation can be used instead of the matrixmultiplication described above to reduce the complexity of the NSST.

In an example, 35×3 non-separable secondary transforms are available forboth 4×4 and 8×8 block sizes, where 35 is a number of transform setsassociated with the intra prediction modes, and 3 is a number of NSSTcandidates for each intra prediction mode. Table 4 shows an exemplarymapping from an intra prediction mode to a respective transform set. Atransform set applied to luma/chroma transform coefficients is specifiedby a corresponding luma/chroma intra prediction mode, according to Table4 that shows mapping from an intra prediction mode to a transform setindex. For an intra prediction mode larger than 34, which corresponds toa diagonal prediction direction, a transform coefficient block istransposed before/after the NSST at the encoder/decoder, respectively.

For each transform set, a selected NSST candidate can be furtherspecified by an explicitly signaled CU level NSST index. The CU levelNSST index is signaled in a bitstream for each intra coded CU aftertransform coefficients and a truncated unary binarization is used forthe CU level NSST index. For example, a truncated value is 2 for theplanar or the DC mode, and 3 for an angular intra prediction mode. In anexample, the CU level NSST index is signaled only when there is morethan one non-zero coefficient in the CU. The default value is zero andnot signaled, indicating that a NSST is not applied to the CU. Each ofvalues 1-3 indicates which NSST candidate is to be applied from thetransform set.

TABLE 4 Intra mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 set 0 1 2 34 5 6 7 8 9 10 11 12 13 14 15 16 Intra mode 17 18 19 20 21 22 23 24 2526 27 28 29 30 31 32 33 set 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3132 33 Intra mode 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 set34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 Intra mode 51 52 5354 55 56 57 58 59 60 61 62 63 64 65 66 67(LM) set 17 16 15 14 13 12 1110 9 8 7 6 5 4 3 2 NULL

In some embodiments, a NSST is not applied for a block coded with atransform skip mode. When the CU level NSST index is signaled for a CUand not equal to zero, a NSST is not used for a block that is coded withthe transform skip mode in the CU. When the CU with blocks of allcomponents are coded in a transform skip mode or a number of non-zerocoefficients of non-transform-skip mode CBs is less than 2, the CU levelNSST index is not signaled for the CU.

Referring to Table 3, when DCT-2 is used in a primary transform, theworst case MPS used for the primary transform can correspond to a TBsize of 32×32 or 32×64. In an example, the worst case MPS uses ⅓ moremultiplications with respect to the second worst case (e.g., a TB sizeof 64×64 or 64×32 when using DCT2). Aspects of the disclosure includemethods that reduce a MPS of a TB size of 32×32 or 32×64 when usingDCT-2 to perform a primary transform, such as an inverse primarytransform. Embodiments herein can also be applied to reduce a MPS of anysuitable TB sizes and using any suitable transform that is not limitedto DCT-2. The suitable transform can use a transform matrix in DCTand/or DST families including DCT-2, DST-7, DST-4, DCT-8, DCT-4, and thelike.

Embodiments described herein may be used separately or combined in anyorder. Further, the embodiments may be implemented by processingcircuitry (e.g., one or more processors or one or more integratedcircuits) in an encoder, a decoder, or the like. In one example, the oneor more processors can execute a program that is stored in anon-transitory computer-readable medium.

In the disclosure, embodiments that are applicable for DST-7 of an MTScandidate can be applicable to DST-4 and vice versa. Similarly,embodiments that are applicable for DCT-8 of an MTS candidate are alsoapplicable to DCT-4 and vice versa.

In the disclosure, embodiments that are applicable for NSST can beapplicable for Reduced Secondary Transform (RST) that is an alternativedesign of NSST. A secondary transform can refer to NSST or RST.

As described above in FIG. 11B, a first order in which an inverseprimary transform is applied on a TB is applying an inverse verticaltransform first followed by an inverse horizontal transform. Accordingto some embodiments, a second order can be used for applying the inverseprimary transform in which the inverse horizontal transform is appliedfirst, and the inverse vertical transform is applied after applying theinverse horizontal transform. Accordingly, the first order can beswitched to the second order. In general, the inverse primary transformcan use any suitable type of primary transforms, such as a transformmatrix in DCT and/or DST families including DCT-2, DST-7, DST-4, DCT-8,DCT-4, and the like. In some embodiments, the inverse primary transformis implemented with a DCT-2 transform matrix. In an example, the inverseprimary transform is implemented with a direct matrix multiplicationstructure. In another example, the inverse primary transform isimplemented with a structure of half butterfly and half direct matrixmultiplication.

In an embodiment, the second order is used (e.g., the first order isswitched to the second order) for certain TB sizes. Accordingly, anorder of applying the inverse horizontal and vertical transforms candepend on a TB size. The TB size can refer to an area of the TB, such asM×N, a width of the TB, a height of the TB, or the like. In an example,when the TB size is 32×64, the second order is used (i.e., the inversehorizontal transform is applied before the inverse vertical transform).When the TB size is different from 32×64, the first order is used (i.e.,the inverse vertical transform is applied before the inverse horizontaltransform).

In an embodiment, when a height N of the TB of M×N is greater than awidth M of the TB, the second order can be used such that the inversehorizontal transform is applied before the inverse vertical transform.Otherwise, when the height N of the TB is smaller than or equal to thewidth M of the TB, the first order can be used such that the inversevertical transform is applied before the inverse horizontal transform.

The inverse horizontal transform can be performed to transform firstelements in a first block of M×N into second elements in a second blockof M×N. The inverse horizontal transform can be referred to as anone-dimensional transform, such as an M-length transform, where theinverse horizontal transform can be applied to the first blockrow-by-row where each row has M first elements. In an embodiment, thesecond elements in left m columns in the second block are calculated bythe inverse horizontal transform while the second elements in right(M-m) columns in the second block are not calculated and are zero. Whenthe inverse horizontal transform includes the zero-out method, m is apositive integer that is less than M, the inverse horizontal transformdescribed above calculates first m second elements in each row of thesecond block while remaining (M-m) second elements in each row are notcalculated and are zero. Thus, the first m second elements (in each row)are kept by the M-length transform using a zero-out method. Accordingly,a number of calculations (and an amount of data to be calculated andstored) can be reduced by including or using the zero-out method. Whenthe inverse horizontal transform does not include the zero-out method, mcan be equal to M.

The inverse vertical transform can be performed to transform thirdelements in a third block of M×N into fourth elements in a fourth blockof M×N. The inverse vertical transform can be referred to as anone-dimensional transform, such as an N-length transform, where theinverse vertical transform can be applied to the third blockcolumn-by-column where each column has N third elements. In anembodiment, the fourth elements in top n rows in the fourth block arecalculated by the inverse vertical transform while the fourth elementsin bottom (N-n) rows in the fourth block are not calculated and arezero. When the inverse vertical transform includes the zero-out method,n is a positive integer that is less than N, the inverse verticaltransform described above calculates first n fourth elements in eachcolumn of the fourth block while remaining (N-n) fourth elements in eachcolumn are not calculated and are zero. Thus, the first n fourthelements (in each column) are kept by the N-length transform using azero-out method. Accordingly, a number of calculations (and an amount ofdata to be calculated and stored) can be reduced by including thezero-out method. When the inverse vertical transform does not includethe zero-out method, n can be equal to N.

The inverse primary transform, e.g., DCT-2, can be applied on acoefficient block having the TB size of M×N to obtain a residual blockof M×N. According to an embodiment of the disclosure, first residualdata in a m×n region in the residual block are to be determined by theinverse primary transform and second residual data outside the m×nregion in the residual block are not to be calculated by the inverseprimary transform where m is less than or equal to M, and n is less thanor equal to N. As described above, when the inverse horizontal transformincludes the zero-out method, m is less than M. When the inversevertical transform includes the zero-out method, n is less than N. Whena ratio m/M is larger than or equal to a ratio n/N, the inversehorizontal transform is applied before applying the inverse verticaltransform. Otherwise, when the ratio m/M is smaller than the ratio n/N,the inverse vertical transform is applied before applying the inversehorizontal transform.

In general, the inverse primary transform is performed to transform thetransform coefficients in the coefficient block into the residual datain the residual block. The TB can refer to the coefficient block and/orthe residual block. The TB size, a size of the coefficient block, and asize of the residual block are M×N. When the ratio m/M is larger than orequal to the ratio n/N, the inverse horizontal transform is appliedbefore applying the inverse vertical transform. Accordingly, the firstblock is a coefficient block of M×N and the first data are transformcoefficients. The second block is an intermediate block of M×N and thesecond data are intermediate data. The third block is identical to thesecond block, and the third data are identical to the second data. Thefourth block is a residual block of M×N and the fourth data are residualdata.

According to some embodiments, an amount of data that is calculated orkept in a one-dimensional transform that is applied to a TB using thezero-out method can be determined based on the TB size. As describedabove, the one-dimensional transform can refer to an inverse horizontaltransform or an inverse vertical transform. In an embodiment, theinverse primary transform is performed using the first order shown inFIG. 11B such that the inverse vertical transform is applied firstfollowed by the inverse horizontal transform. When the TB size is 32×64with a width of 32 and a height of 64, a 64-length inverse verticaltransform can be applied with the zero-out to transform a coefficientblock of 32×64 having transform coefficients into an intermediate blockhaving intermediate data by calculating the top 16 rows of theintermediate block (i.e., first 16 pieces of intermediate data in eachcolumn). Remaining intermediate data in the intermediate block are notcalculated and are zero (e.g., the remaining intermediate data can beset to zero). In some examples, when the TB size is different from32×64, such as 16×64, the 64-length inverse vertical transformcalculates the top 32 rows of the intermediate data in the intermediateblock, and the remaining intermediate data in the intermediate block arenot calculated and are zero.

When the TB size is 32×64 and the first order is used, a 32-lengthinverse horizontal transform can be applied with the zero-out totransform an intermediate block of 32×64 having intermediate data into aresidual block having residual data by calculating the left 16 columnsof the residual block (i.e., first 16 pieces of residual data in eachrow). Remaining residual data in the residual block are not calculatedand are zero (e.g., the remaining residual data can be set to zero).

In some examples, when the TB size is M×32 where M is smaller than 32,the 32-length inverse vertical transform does not apply the zero-outscheme. In some examples, when the TB size is 32×N where N is smallerthan 32, the 32-length inverse horizontal transform does not apply thezero-out scheme.

According to some embodiments, an amount of data to be calculated for arow or a column using a one-dimensional transform can be determinedbased on whether the one-dimensional transform is an inverse horizontaltransform or an inverse vertical transform. For example, a zero-outscheme can be used for only one of the one-dimensional transforms. In anembodiment, the inverse primary transform is performed using the firstorder such that the inverse vertical transform is applied first followedby the inverse horizontal transform. The TB size is 32×32. In anexample, the 32-length vertical transform does not apply the zero-outmethod. Accordingly, the 32-length inverse vertical transform calculatesor keeps all intermediate data in 32 rows of an intermediate block. The32-length inverse horizontal transform calculates or keeps the left 16columns of a residual block using the zero-out method. Remainingresidual data in the residual block are not calculated and are zero(e.g., the remaining residual data can be set to zero). In an example,the 32-length inverse vertical transform calculates or keeps the top 16rows of an intermediate block using the zero-out method. Remainingintermediate data in the intermediate block are not calculated and arezero (e.g., the remaining intermediate data can be set to zero).Meanwhile, the 32-length inverse horizontal transform does not apply thezero-out method. Accordingly, the 32-length inverse horizontal transformcalculates or keeps all residual data in 32 columns of a residual block.

When a first inverse primary transform is applied to a TB without aninverse secondary transform, a first MPS is due to the first inverseprimary transform. When a second inverse primary transform and aninverse secondary transform are applied to the TB, a second MPS (alsoreferred to as a combined MPS) can include multiplications in both thesecond inverse primary transform and the inverse secondary transform.When the first inverse primary transform is identical to the secondinverse primary transform, the second MPS can be larger than the firstMPS because additional multiplications are used in the inverse secondarytransform. Embodiments below can be implemented to constrain or reducethe combined MPS to be within a MPS limit (e.g., the worse case MPSshown in Table 3) and thus to improve coding efficiency.

According to aspects of the disclosure, a number of non-zero transformcoefficients that are kept using a zero-out scheme, and/or whether azero-out method is used in an inverse primary transform, can depend onwhether a secondary transform is applied. Thus, a combined MPS for theinverse primary and secondary transforms can be reduced. Referring backto the first and second inverse primary transforms, according to anaspect of the disclosure, the first inverse primary transform maycalculate or keep a first number of residual data, and the secondinverse primary transform may calculate or keep a second number ofresidual data where the second number of residual data is less than thefirst number of residual data so that a combined MPS (i.e., the secondMPS) is similar to the 1^(st) MPS. In an example, at least oneone-dimensional transform (e.g., an inverse horizontal transform or aninverse vertical transform) of the second inverse primary transformincludes the zero-out method. A one-dimensional transform of the firstinverse primary transform can also include the zero-out method, however,the second inverse primary transform can calculate or keep less residualdata than those of the first inverse primary transform.

As described above, an inverse primary transform and an inversesecondary transform can be applied to a TB and result in a combined MPS.In general, inverse primary transforms can include different types(e.g., DCT-2, DST-7, DST-4, DCT-8, DCT-4, or the like). According toaspects of the disclosure, whether to use the zero-out method and/or anamount of data to be calculated or kept in a coefficient block whenusing the zero-out method in the inverse secondary transform can bedetermined based on a type of the inverse primary transform, and thusthe combined MPS can be reduced. For example, when the inverse primarytransform is more complex (e.g., a MPS of the inverse primary transformis relatively large), an amount of data to be calculated or kept in thecoefficient block when using the zero-out method in the inversesecondary transform can be further reduced.

According to aspects of the disclosure, the zero-out method can beperformed depending on an intra prediction mode used for the TB. In someembodiments, whether to use the zero-out method and an amount of data tobe calculated or kept when using the zero-out method can be determinedbased on the intra prediction mode.

In an embodiment, whether an inverse horizontal transform or an inversevertical transform is performed with the zero-out method can depend onan intra prediction direction used for the TB. The intra predictiondirection corresponds to the intra prediction mode. In an example, whenthe intra prediction direction is close to a horizontal predictiondirection, the zero-out method is applied for the inverse horizontaltransform. In an example, when the intra prediction direction is closeto a vertical prediction direction, the zero-out method is applied forthe inverse vertical transform. For example, the horizontal predictiondirection is indexed by a value Hor, and the vertical predictiondirection is indexed by a value Ver. A threshold thres can be defined todetermine a closesness of the intra prediction mode to a horizontal modecorresponding to the the horizontal prediction direction or a verticalmode corresponding to the the vertical prediction direction. The intraprediction mode (or the intra prediction direction) indexed within arange of [Hor−thres, Hor+thres] is determined to be close to thehorizontal prediction direction. The intra prediction mode indexedwithin a range of [Ver−thres, Ver+thres] can be determined to be closeto the vertical prediction direction. Example values of the thresholdthres include, but are not limited to 1, 2, 3, 4, . . . , and 16.

In an embodiment, an amount of data to be calculated or kept when usingthe zero-out method in the inverse primary transform and/or secondarytransform can be determined based on the intra prediction direction forthe TB.

FIG. 14 shows a flow chart outlining a process (1400) according to anembodiment of the disclosure. The process (1400) can be used in thereconstruction of a block coded in intra mode, so to generate aprediction block for the block under reconstruction. In some examples,the process (1400) can be used in the reconstruction of a block coded ininter mode. In various embodiments, the process (1400) are executed byprocessing circuitry, such as the processing circuitry in the terminaldevices (310), (320), (330) and (340), the processing circuitry thatperforms functions of the video encoder (403), the processing circuitrythat performs functions of the video decoder (410), the processingcircuitry that performs functions of the video decoder (510), theprocessing circuitry that performs functions of the video encoder (603),and the like. In some embodiments, the process (1400) is implemented insoftware instructions, thus when the processing circuitry executes thesoftware instructions, the processing circuitry performs the process(1400). The process starts at (S1401) and proceeds to (S1410).

At (S1410), coding information of a TB, such as a coefficient block,from a coded video bitstream can be decoded. The coding information canindicate a TB size (or referred to as a size of the coefficient block).As described above, the TB size can refer to an area, a width, a height,or the like of the TB.

At (S1420), an order in which inverse horizontal and inverse verticaltransforms of an inverse primary transform are to be performed can bedetermined based on the size of the coefficient block. Transformcoefficients of the coefficient block can be transformed into residualdata of a residual block by the inverse primary transform. As describedabove, in an example, the order is the first order. In an example, theorder is the second order.

At (S1430), whether the TB size satisfies a condition can be determined,as described above. The condition can be whether the TB size is 32×64.The condition can be a height N of the coefficient block of M×N islarger than a width M of the coefficient block. The condition cancompare a first ratio and a second ratio where the first ratio (e.g.,m/M) is an amount of data calculated for a row over a total amount ofdata in the row and the second ratio (n/N) is an amount of datacalculated for a column over a total amount of data in the column. Whenthe TB size is determined to satisfy the condition, the process (1400)proceeds to (S1440). Otherwise, the process (1400) proceeds to (S1450).

At (S1440), the inverse vertical transform is performed after theinverse horizontal transform is performed on the transform coefficientsof the coefficient block, and thus the second order is used.

At (S1450), the inverse horizontal transform is performed after theinverse vertical transform is performed on the transform coefficients ofthe coefficient block, and thus the first order is used.

At (S1460), a sample in the residual block can be reconstructed based onthe residual data.

The process (1400) can be suitably adapted. For example, one or moresteps can be modified, omitted, or combined. For example, (S1420) and(S1430) can be combined. Additional step(s) can also be added. An orderthat the process (1400) is executed can also be modified.

FIG. 15 shows a flow chart outlining a process (1500) according to anembodiment of the disclosure. The process (1500) can be used in thereconstruction of a block coded in intra mode, so to generate aprediction block for the block under reconstruction. In some examples,the process (1500) can be used in the reconstruction of a block coded ininter mode. In various embodiments, the process (1500) are executed byprocessing circuitry, such as the processing circuitry in the terminaldevices (310), (320), (330) and (340), the processing circuitry thatperforms functions of the video encoder (403), the processing circuitrythat performs functions of the video decoder (410), the processingcircuitry that performs functions of the video decoder (510), theprocessing circuitry that performs functions of the video encoder (603),and the like. In some embodiments, the process (1500) is implemented insoftware instructions, thus when the processing circuitry executes thesoftware instructions, the processing circuitry performs the process(1500). The process starts at (S1501) and proceeds to (S1510).

At (S1510), coding information of a TB, such as a coefficient block,from a coded video bitstream can be decoded. The coding information canindicate a TB size (or referred to as a size of the coefficient block).

At (S1520), whether to reduce a number of calculations (or an amount ofdata to be calculated) in one of inverse horizontal and inverse verticaltransforms of an inverse primary transform can be determined based onthe size of the coefficient block. Transform coefficients of thecoefficient block can be transformed into residual data of a residualblock by the inverse primary transform. In an embodiment, the firstorder is used. The inverse vertical transform can transform thetransform coefficients of the coefficient block to intermediate data ofan intermediate block, and the inverse horizontal transform cantransform the intermediate data to the residual data in the residualblock. In an example, reducing the number of calculations in theone-dimensional transform (e.g., the inverse horizontal transform or theinverse vertical transform) means the one-dimensional transform includesthe zero-out method where first 16 rows (for the inverse horizontaltransform) or columns (for the inverse vertical transform) of data arecalculated and remaining data are not calculated. For example, when theTB size is one of 32×64 and 32×32, the number of calculations in theone-dimensional transform is determined to be reduced. When the numberof calculations in the one-dimensional transform is determined to bereduced, the process (1500) proceeds to (S1530).

At (S1530), whether the number of calculations in the inverse horizontaltransform is to be reduced can be determined. When the number ofcalculations in the inverse horizontal transform is to be reduced, theprocess (1500) proceeds to (S1540). Otherwise, the process (1500)proceeds to (S1550).

(S1530) can be suitably modified. For example, instead of determiningwhether the number of calculations in the inverse horizontal transformis to be reduced, at (S1530), whether the number of calculations in theinverse vertical transform is to be reduced can be determined.

At (S1540), the inverse primary transform is performed where the inversehorizontal transform includes the zero-out method, and left 16 columnsof the residual data in the residual block are calculated by the inversehorizontal transform and the remaining residual data in the residualblock are not calculated and are zero, as described above.

In an example, at (S1540), the inverse vertical transform also includesthe zero-out method.

At (S1550), the inverse primary transform is performed where the inversevertical transform includes the zero-out method, and thus top 16 rows ofthe intermediate data in the intermediate block are calculated by theinverse vertical transform and the remaining intermediate data in theintermediate block are not calculated and are zero, as described above.

At (S1560), a sample in the residual block can be reconstructed based onthe residual data.

The process (1500) can be suitably adapted. For example, one or moresteps can be modified, omitted, or combined. For example, steps (S1520)and (S1530) can be combined into a single step. Additional step(s) canalso be added. An order that the process (1500) is executed can also bemodified.

In an example, step(s) in the process (1400) and step(s) in the process(1500) can be modified and combined.

The techniques described above, can be implemented as computer softwareusing computer-readable instructions and physically stored in one ormore computer-readable media. For example, FIG. 16 shows a computersystem (1600) suitable for implementing certain embodiments of thedisclosed subject matter.

The computer software can be coded using any suitable machine code orcomputer language, that may be subject to assembly, compilation,linking, or like mechanisms to create code comprising instructions thatcan be executed directly, or through interpretation, micro-codeexecution, and the like, by one or more computer central processingunits (CPUs), Graphics Processing Units (GPUs), and the like.

The instructions can be executed on various types of computers orcomponents thereof, including, for example, personal computers, tabletcomputers, servers, smartphones, gaming devices, internet of thingsdevices, and the like.

The components shown in FIG. 16 for computer system (1600) are exemplaryin nature and are not intended to suggest any limitation as to the scopeof use or functionality of the computer software implementingembodiments of the present disclosure. Neither should the configurationof components be interpreted as having any dependency or requirementrelating to any one or combination of components illustrated in theexemplary embodiment of a computer system (1600).

Computer system (1600) may include certain human interface inputdevices. Such a human interface input device may be responsive to inputby one or more human users through, for example, tactile input (such as:keystrokes, swipes, data glove movements), audio input (such as: voice,clapping), visual input (such as: gestures), olfactory input (notdepicted). The human interface devices can also be used to capturecertain media not necessarily directly related to conscious input by ahuman, such as audio (such as: speech, music, ambient sound), images(such as: scanned images, photographic images obtain from a still imagecamera), video (such as two-dimensional video, three-dimensional videoincluding stereoscopic video).

Input human interface devices may include one or more of (only one ofeach depicted): keyboard (1601), mouse (1602), trackpad (1603), touchscreen (1610), data-glove (not shown), joystick (1605), microphone(1606), scanner (1607), camera (1608).

Computer system (1600) may also include certain human interface outputdevices. Such human interface output devices may be stimulating thesenses of one or more human users through, for example, tactile output,sound, light, and smell/taste. Such human interface output devices mayinclude tactile output devices (for example tactile feedback by thetouch-screen (1610), data-glove (not shown), or joystick (1605), butthere can also be tactile feedback devices that do not serve as inputdevices), audio output devices (such as: speakers (1609), headphones(not depicted)), visual output devices (such as screens (1610) toinclude CRT screens, LCD screens, plasma screens, OLED screens, eachwith or without touch-screen input capability, each with or withouttactile feedback capability—some of which may be capable to output twodimensional visual output or more than three dimensional output throughmeans such as stereographic output; virtual-reality glasses (notdepicted), holographic displays and smoke tanks (not depicted)), andprinters (not depicted).

Computer system (1600) can also include human accessible storage devicesand their associated media such as optical media including CD/DVD ROM/RW(1620) with CD/DVD or the like media (1621), thumb-drive (1622),removable hard drive or solid state drive (1623), legacy magnetic mediasuch as tape and floppy disc (not depicted), specialized ROM/ASIC/PLDbased devices such as security dongles (not depicted), and the like.

Those skilled in the art should also understand that term “computerreadable media” as used in connection with the presently disclosedsubject matter does not encompass transmission media, carrier waves, orother transitory signals.

Computer system (1600) can also include an interface to one or morecommunication networks. Networks can for example be wireless, wireline,optical. Networks can further be local, wide-area, metropolitan,vehicular and industrial, real-time, delay-tolerant, and so on. Examplesof networks include local area networks such as Ethernet, wireless LANs,cellular networks to include GSM, 3G, 4G, 5G, LTE and the like, TVwireline or wireless wide area digital networks to include cable TV,satellite TV, and terrestrial broadcast TV, vehicular and industrial toinclude CANBus, and so forth. Certain networks commonly require externalnetwork interface adapters that attached to certain general purpose dataports or peripheral buses (1649) (such as, for example USB ports of thecomputer system (1600)); others are commonly integrated into the core ofthe computer system (1600) by attachment to a system bus as describedbelow (for example Ethernet interface into a PC computer system orcellular network interface into a smartphone computer system). Using anyof these networks, computer system (1600) can communicate with otherentities. Such communication can be uni-directional, receive only (forexample, broadcast TV), uni-directional send-only (for example CANbus tocertain CANbus devices), or bi-directional, for example to othercomputer systems using local or wide area digital networks. Certainprotocols and protocol stacks can be used on each of those networks andnetwork interfaces as described above.

Aforementioned human interface devices, human-accessible storagedevices, and network interfaces can be attached to a core (1640) of thecomputer system (1600).

The core (1640) can include one or more Central Processing Units (CPU)(1641), Graphics Processing Units (GPU) (1642), specialized programmableprocessing units in the form of Field Programmable Gate Areas (FPGA)(1643), hardware accelerators for certain tasks (1644), and so forth.These devices, along with Read-only memory (ROM) (1645), Random-accessmemory (1646), internal mass storage such as internal non-useraccessible hard drives, SSDs, and the like (1647), may be connectedthrough a system bus (1648). In some computer systems, the system bus(1648) can be accessible in the form of one or more physical plugs toenable extensions by additional CPUs, GPU, and the like. The peripheraldevices can be attached either directly to the core's system bus (1648),or through a peripheral bus (1649). Architectures for a peripheral businclude PCI, USB, and the like.

CPUs (1641), GPUs (1642), FPGAs (1643), and accelerators (1644) canexecute certain instructions that, in combination, can make up theaforementioned computer code. That computer code can be stored in ROM(1645) or RAM (1646). Transitional data can be also be stored in RAM(1646), whereas permanent data can be stored for example, in theinternal mass storage (1647). Fast storage and retrieve to any of thememory devices can be enabled through the use of cache memory, that canbe closely associated with one or more CPU (1641), GPU (1642), massstorage (1647), ROM (1645), RAM (1646), and the like.

The computer readable media can have computer code thereon forperforming various computer-implemented operations. The media andcomputer code can be those specially designed and constructed for thepurposes of the present disclosure, or they can be of the kind wellknown and available to those having skill in the computer software arts.

As an example and not by way of limitation, the computer system havingarchitecture (1600), and specifically the core (1640) can providefunctionality as a result of processor(s) (including CPUs, GPUs, FPGA,accelerators, and the like) executing software embodied in one or moretangible, computer-readable media. Such computer-readable media can bemedia associated with user-accessible mass storage as introduced above,as well as certain storage of the core (1640) that are of non-transitorynature, such as core-internal mass storage (1647) or ROM (1645). Thesoftware implementing various embodiments of the present disclosure canbe stored in such devices and executed by core (1640). Acomputer-readable medium can include one or more memory devices orchips, according to particular needs. The software can cause the core(1640) and specifically the processors therein (including CPU, GPU,FPGA, and the like) to execute particular processes or particular partsof particular processes described herein, including defining datastructures stored in RAM (1646) and modifying such data structuresaccording to the processes defined by the software. In addition or as analternative, the computer system can provide functionality as a resultof logic hardwired or otherwise embodied in a circuit (for example:accelerator (1644)), which can operate in place of or together withsoftware to execute particular processes or particular parts ofparticular processes described herein. Reference to software canencompass logic, and vice versa, where appropriate. Reference to acomputer-readable media can encompass a circuit (such as an integratedcircuit (IC)) storing software for execution, a circuit embodying logicfor execution, or both, where appropriate. The present disclosureencompasses any suitable combination of hardware and software.

-   Appendix A: Acronyms-   JEM: joint exploration model-   VVC: versatile video coding-   BMS: benchmark set-   MV: Motion Vector-   HEVC: High Efficiency Video Coding-   SEI: Supplementary Enhancement Information-   VUI: Video Usability Information-   GOPs: Groups of Pictures-   TUs: Transform Units,-   PUs: Prediction Units-   CTUs: Coding Tree Units-   CTBs: Coding Tree Blocks-   PBs: Prediction Blocks-   HRD: Hypothetical Reference Decoder-   SNR: Signal Noise Ratio-   CPUs: Central Processing Units-   GPUs: Graphics Processing Units-   CRT: Cathode Ray Tube-   LCD: Liquid-Crystal Display-   OLED: Organic Light-Emitting Diode-   CD: Compact Disc-   DVD: Digital Video Disc-   ROM: Read-Only Memory-   RAM: Random Access Memory-   ASIC: Application-Specific Integrated Circuit-   PLD: Programmable Logic Device-   LAN: Local Area Network-   GSM: Global System for Mobile communications-   LTE: Long-Term Evolution-   CANBus: Controller Area Network Bus-   USB: Universal Serial Bus-   PCI: Peripheral Component Interconnect-   FPGA: Field Programmable Gate Areas-   SSD: solid-state drive-   IC: Integrated Circuit-   CU: Coding Unit

While this disclosure has described several exemplary embodiments, thereare alterations, permutations, and various substitute equivalents, whichfall within the scope of the disclosure. It will thus be appreciatedthat those skilled in the art will be able to devise numerous systemsand methods which, although not explicitly shown or described herein,embody the principles of the disclosure and are thus within the spiritand scope thereof.

APPENDIX I 4x4 transform {64, 64, 64, 64} {83, 36, −36, −83} {64, −64,−64, 64} {36, −83, 83, −36} 8x8 transform {64, 64, 64, 64, 64, 64, 64,64} {89, 75, 50, 18, −18, −50, −75, −89} {83, 36, −36, −83, −83, −36,36, 83} {75, −18, −89, −50, 50, 89, 18, −75} {64, −64, −64, 64, 64, −64,−64, 64} {50, −89, 18, 75, −75, −18, 89, −50} {36, −83, 83, −36, −36,83, −83, 36} {18, −50, 75, −89, 89, −75, 50, −18} 16x16 transform {64 6464 64 64 64 64 64 64 64 64 64 64 64 64 64} {90 87 80 70 57 43 25 9 −9−25 −43 −57 −70 −80 −87 −90} {89 75 50 18 −18 −50 −75 −89 −89 −75 −50−18 18 50 75 89} {87 57 9 −43 −80 −90 −70 −25 25 70 90 80 43 −9 −57 −87}{83 36 −36 −83 −83 −36 36 83 83 36 −36 −83 −83 −36 36 83} {80 9 −70 −87−25 57 90 43 −43 −90 −57 25 87 70 −9 −80} {75 −18 −89 −50 50 89 18 −75−75 18 89 50 −50 −89 −18 75} {70 −43 −87 9 90 25 −80 −57 57 80 −25 −90−9 87 43 −70} {64 −64 −64 64 64 −64 −64 64 64 −64 −64 64 64 −64 −64 64}{57 −80 −25 90 −9 −87 43 70 −70 −43 87 9 −90 25 80 −57} {50 −89 18 75−75 −18 89 −50 −50 89 −18 −75 75 18 −89 50} { 43 −90 57 25 −87 70 9 −8080 −9 −70 87 −25 −57 90 −43} {36 −83 83 −36 −36 83 −83 36 36 −83 83 −36−36 83 −83 36} {25 −70 90 −80 43 9 −57 87 −87 57 −9 −43 80 −90 70 −25}{18 −50 75 −89 89 −75 50 −18 −18 50 −75 89 −89 75 −50 18} {9 −25 43 −5770 −80 87 −90 90 −87 80 −70 57 −43 25 −9} 32x32 transform {64 64 64 6464 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 6464 64 64 64} {90 90 88 85 82 78 73 67 61 54 46 38 31 22 13 4 −4 −13 −22−31 −38 −46 −54 −61 −67 −73 −78 −82 −85 −88 −90 −90} {90 87 80 70 57 4325 9 −9 −25 −43 −57 −70 −80 −87 −90 −90 −87 −80 −70 −57 −43 −25 −9 9 2543 57 70 80 87 90} {90 82 67 46 22 −4 −31 −54 −73 −85 −90 −88 −78 −61−38 −13 13 38 61 78 88 90 85 73 54 31 4 −22 −46 −67 −82 −90} {89 75 5018 −18 −50 −75 −89 −89 −75 −50 −18 18 50 75 89 89 75 50 18 −18 −50 −75−89 −89 −75 −50 −18 18 50 75 89} {88 67 31 −13 −54 −82 −90 −78 −46 −4 3873 90 85 61 22 −22 −61 −85 −90 −73 −38 4 46 78 90 82 54 13 −31 −67 −88}{87 57 9 −43 −80 −90 −70 −25 25 70 90 80 43 −9 −57 −87 −87 −57 −9 43 8090 70 25 −25 −70 −90 −80 −43 9 57 87} {85 46 −13 −67 −90 −73 −22 38 8288 54 −4 −61 −90 −78 −31 31 78 90 61 4 −54 −88 −82 −38 22 73 90 67 13−46 −85} {83 36 −36 −83 −83 −36 36 83 83 36 −36 −83 −83 −36 36 83 83 36−36 −83 −83 −36 36 83 83 36 −36 −83 −83 −36 36 83} {82 22 −54 −90 −61 1378 85 31 −46 −90 −67 4 73 88 38 −38 −88 −73 −4 67 90 46 −31 −85 −78 −1361 90 54 −22 −82} {80 9 −70 −87 −25 57 90 43 −43 −90 −57 25 87 70 −9 −80−80 −9 70 87 25 −57 −90 −43 43 90 57 −25 −87 −70 9 80} {78 −4 −82 −73 1385 67 −22 −88 −61 31 90 54 −38 −90 −46 46 90 38 −54 −90 −31 61 88 22 −67−85 −13 73 82 4 −78} {75 −18 −89 −50 50 89 18 −75 −75 18 89 50 −50 −89−18 75 75 −18 −89 −50 50 89 18 −75 −75 18 89 50 −50 −89 −18 75} {73 −31−90 −22 78 67 −38 −90 −13 82 61 −46 −88 −4 85 54 −54 −85 4 88 46 −61 −8213 90 38 −67 −78 22 90 31 −73} {70 −43 −87 9 90 25 −80 −57 57 80 −25 −90−9 87 43 −70 −70 43 87 −9 −90 −25 80 57 −57 −80 25 90 9 −87 −43 70} {67−54 −78 38 85 −22 −90 4 90 13 −88 −31 82 46 −73 −61 61 73 −46 −82 31 88−13 −90 −4 90 22 −85 −38 78 54 −67} {64 −64 −64 64 64 −64 −64 64 64 −64−64 64 64 −64 −64 64 64 −64 −64 64 64 −64 −64 64 64 −64 −64 64 64 −64−64 64} {61 −73 −46 82 31 −88 −13 90 −4 −90 22 85 −38 −78 54 67 −67 −5478 38 −85 −22 90 4 −90 13 88 −31 −82 46 73 −61} {57 −80 −25 90 −9 −87 4370 −70 −43 87 9 −90 25 80 −57 −57 80 25 −90 9 87 −43 −70 70 43 −87 −9 90−25 −80 57} {54 −85 −4 88 −46 −61 82 13 −90 38 67 −78 −22 90 −31 −73 7331 −90 22 78 −67 −38 90 −13 −82 61 46 −88 4 85 −54} {50 −89 18 75 −75−18 89 −50 −50 89 −18 −75 75 18 −89 50 50 −89 18 75 −75 −18 89 −50 −5089 −18 −75 75 18 −89 50} {46 −90 38 54 −90 31 61 −88 22 67 −85 13 73 −824 78 −78 −4 82 −73 −13 85 −67 −22 88 −61 −31 90 −54 −38 90 −46} { 43 −9057 25 −87 70 9 −80 80 −9 −70 87 −25 −57 90 −43 −43 90 −57 −25 87 −70 −980 −80 9 70 −87 25 57 −90 43} {38 −88 73 −4 −67 90 −46 −31 85 −78 13 61−90 54 22 −82 82 −22 −54 90 −61 −13 78 −85 31 46 −90 67 4 −73 88 −38}{36 −83 83 −36 −36 83 −83 36 36 −83 83 −36 −36 83 −83 36 36 −83 83 −36−36 83 −83 36 36 −83 83 −36 −36 83 −83 36} {31 −78 90 −61 4 54 −88 82−38 −22 73 −90 67 −13 −46 85 −85 46 13 −67 90 −73 22 38 −82 88 −54 −4 61−90 78 −31} {25 −70 90 −80 43 9 −57 87 −87 57 −9 −43 80 −90 70 −25 −2570 −90 80 −43 −9 57 −87 87 −57 9 43 −80 90 −70 25} {22 −61 85 −90 73 −38−4 46 −78 90 −82 54 −13 −31 67 −88 88 −67 31 13 −54 82 −90 78 −46 4 38−73 90 −85 61 −22} {18 −50 75 −89 89 −75 50 −18 −18 50 −75 89 −89 75 −5018 18 −50 75 −89 89 −75 50 −18 −18 50 −75 89 −89 75 −50 18} {13 −38 61−78 88 −90 85 −73 54 −31 4 22 −46 67 −82 90 −90 82 −67 46 −22 −4 31 −5473 −85 90 −88 78 −61 38 −13} {9 −25 43 −57 70 −80 87 −90 90 −87 80 −7057 −43 25 −9 −9 25 −43 57 −70 80 −87 90 −90 87 −80 70 −57 43 −25 9} {4−13 22 −31 38 −46 54 −61 67 −73 78 −82 85 −88 90 −90 90 −90 88 −85 82−78 73 −67 61 −54 46 −38 31 −22 13 −4}

APPENDIX II 64-point DCT-2 core {  { aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa }  { bf, bg, bh, bi, bj, bk, bi, bm, bn, bo, bp, bq, br, bs, bt, bu,by, bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, -ck,-cj, -ci, -ch, -cg, -cf, -ce, -cd, -cc, -cb, -ca, -bz, -by, -bx, -bw,-by, -bu, -bt, -bs, -br, -bq, -bp, -bo, -bn, -bm, -bi, -bk, -bj, -bi,-bh, -bg, -bf }  { ap, aq, ar, as, at, au, ay, aw, ax, ay, az, ba, bb,bc, bd, be, -be, -bd, -bc, -bb, -ba, -az, -ay, -ax, -aw, -ay, -au, -at,-as, -ar, -aq, -ap, -ap, -aq, -ar, -as, -at, -au, -ay, -aw, -ax, -ay,-az, -ba, -bb, -bc, -bd, -be, be, bd, bc, bb, ba, az, ay, ax, aw, ay,au, at, as, ar, aq, ap }  { bg, bj, bm, bp, bs, by, by, cb, ce, ch, ck,-ci, -cf, -cc, -bz, -bw, -bt, -bq, -bn, -bk, -bh, -bf, -bi, -bi, -bo,-br, -bu, -bx, -ca, -cd, -cg, -cj, cj, cg, cd, ca, bx, bu, br, bo, bi,bi, bf, bh, bk, bn, bq, bt, bw, bz, cc, cf, ci, -ck, -ch, -ce, -cb, -by,-by, -bs, -bp, -bm, -bj, -bg }  { ah, ai, aj, ak, al, am, an, ao, -ao,-an, -am, -al, -ak, -aj, -ai, -ah, -ah, -ai, -aj, -ak, -al, -am, -an,-ao, ao, an, am, al, ak, aj, ai, ah, ah, ai, aj, ak, al, am, an, ao,-ao, -an, -am, -al, -ak, -aj, -ai, -ah, -ah, -ai, -aj, -ak, -al, -am,-an, -ao, ao, an, am, al, ak, aj, ai, ah }  { bh, bm, br, bw, cb, cg,-ck, -cf, -ca, -by, -bq, -bl, -bg, -bi, -bn, -bs, -bx, -cc, -ch, cj, ce,bz, bu, bp, bk, bf, bj, bo, bt, by, cd, ci, -ci, -cd, -by, -bt, -bo,-bj, -bf, -bk, -bp, -bu, -bz, -ce, -cj, ch, cc, bx, bs, bn, bi, bg, bl,bq, by, ca, cf, ck, -cg, -cb, -bw, -br, -bm, -bh }  { aq, at, aw, az,bc, -be, -bb, -ay, -ay, -as, -ap, -ar, -au, -ax, -ba, -bd, bd, ba, ax,au, ar, ap, as, ay, ay, bb, be, -bc, -az, -aw, -at, -aq, -aq, -at, -aw,-az, -bc, be, bb, ay, ay, as, ap, ar, au, ax, ba, bd, -bd, -ba, -ax,-au, -ar, -ap, -as, -ay, -ay, -bb, -be, bc, az, aw, at, aq }  { bi, bp,bw, cd, ck, -ce, -bx, -bq, -bj, -bh, -bo, -by, -cc, -cj, cf, by, br, bk,bg, bn, bu, cb, ci, -cg, -bz, -bs, -bi, -bf, -bm, -bt, -ca, -ch, ch, ca,bt, bm, bf, bi, bs, bz, cg, -ci, -cb, -bu, -bn, -bg, -bk, -br, -by, -cf,cj, cc, by, bo, bh, bj, bq, bx, ce, -ck, -cd, -bw, -bp, -bi }  { ad, ae,af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad, ad, ae,af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad, ad, ae,af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad, ad, ae,af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad }  { bj,bs, cb, ck, -cc, -bt, -bk, -bi, -br, -ca, -cj, cd, bu, bi, bh, bq, bz,ci, -ce, -by, -bm, -bg, -bp, -by, -ch, cf, bw, bn, bf, bo, bx, cg, -cg,-bx, -bo, -bf, -bn, -bw, -cf, ch, by, bp, bg, bm, by, ce, -ci, -bz, -bq,-bh, -bi, -bu, -cd, cj, ca, br, bi, bk, bt, cc, -ck, -cb, -bs, -bj }  {ar, aw, bb, -bd, -ay, -at, -ap, -au, -az, -be, ba, ay, aq, as, ax, bc,-bc, -ax, -as, -aq, -ay, -ba, be, az, au, ap, at, ay, bd, -bb, -aw, -ar,-ar, -aw, -bb, bd, ay, at, ap, au, az, be, -ba, -ay, -aq, -as, -ax, -bc,bc, ax, as, aq, ay, ba, -be, -az, -au, -ap, -at, -ay, -bd, bb, aw, ar } { bk, by, cg, -ce, -bt, -bi, -bm, -bx, -ci, cc, br, bg, bo, bz, ck,-ca, -bp, -bf, -bq, -cb, cj, by, bn, bh, bs, cd, -ch, -bw, -bi, -bj,-bu, -cf, cf, bu, bj, bi, bw, ch, -cd, -bs, -bh, -bn, -by, -cj, cb, bq,bf, bp, ca, -ck, -bz, -bo, -bg, -br, -cc, ci, bx, bm, bi, bt, ce, -cg,-by, -bk }  { ai, al, ao, -am, -aj, -ah, -ak, -an, an, ak, ah, aj, am,-ao, -al, -ai, -ai, -al, -ao, am, aj, ah, ak, an, -an, -ak, -ah, -aj,-am, ao, al, ai, ai, al, ao, -am, -aj, -ah, -ak, -an, an, ak, ah, aj,am, -ao, -al, -ai, -ai, -al, -ao, am, aj, ah, ak, an, -an, -ak, -ah,-aj, -am, ao, al, ai }  { bl, by, -ck, -bx, -bk, -bm, -bz, cj, bw, bj,bn, ca, -ci, -by, -bi, -bo, -cb, ch, bu, bh, bp, cc, -cg, -bt, -bg, -bq,-cd, cf, bs, bf, br, ce, -ce, -br, -bf, -bs, -cf, cd, bq, bg, bt, cg,-cc, -bp, -bh, -bu, -ch, cb, bo, bi, by, ci, -ca, -bn, -bj, -bw, -cj,bz, bm, bk, bx, ck, -by, -bl }  { as, az, -bd, -aw, -ap, -ay, -bc, ba,at, ar, ay, -be, -ax, -aq, -au, -bb, bb, au, aq, ax, be, -ay, -ar, -at,-ba, bc, ay, ap, aw, bd, -az, -as, -as, -az, bd, aw, ap, ay, bc, -ba,-at, -ar, -ay, be, ax, aq, au, bb, -bb, -au, -aq, -ax, -be, ay, ar, at,ba, -bc, -ay, -ap, -aw, -bd, az, as }  { bm, cb, -cf, -bq, -bi, -bx, cj,bu, bf, bt, ci, -by, -bj, -bp, -ce, cc, bn, bi, ca, -cg, -br, -bh, -bw,ck, by, bg, bs, ch, -bz, -bk, -bo, -cd, cd, bo, bk, bz, -ch, -bs, -bg,-by, -ck, bw, bh, br, cg, -ca, -bi, -bn, -cc, ce, bp, bj, by, -ci, -bt,-bf, -bu, -cj, bx, bi, bq, cf, -cb, -bm }  { ab, ac, -ac, -ab, -ab, -ac,ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac,ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac,ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac,ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab }  { bn, ce, -ca, -bj, -br,-ci, bw, bf, by, -cj, -bs, -bi, -bz, cf, bo, bm, cd, -cb, -bk, -bq, -ch,bx, bg, bu, -ck, -bt, -bh, -by, cg, bp, bi, cc, -cc, -bi, -bp, -cg, by,bh, bt, ck, -bu, -bg, -bx, ch, bq, bk, cb, -cd, -bm, -bo, -cf, bz, bi,bs, cj, -by, -bf, -bw, ci, br, bj, ca, -ce, -bn }  { at, bc, -ay, -ap,-ax, bd, au, as, bb, -az, -aq, -aw, be, ay, ar, ba, -ba, -ar, -ay, -be,aw, aq, az, -bb, -as, -au, -bd, ax, ap, ay, -bc, -at, -at, -bc, ay, ap,ax, -bd, -au, -as, -bb, az, aq, aw, -be, -ay, -ar, -ba, ba, ar, ay, be,-aw, -aq, -az, bb, as, au, bd, -ax, -ap, -ay, bc, at }  { bo, ch, -by,-bh, -ca, cc, bj, bt, -cj, -bq, -bm, -cf, bx, bf, by, -ce, -bi, -br,-ck, bs, bk, cd, -bz, -bg, -bw, cg, bn, bp, ci, -bu, -bi, -cb, cb, bi,bu, -ci, -bp, -bn, -cg, bw, bg, bz, -cd, -bk, -bs, ck, br, bi, ce, -by,-bf, -bx, cf, bm, bq, cj, -bt, -bj, -cc, ca, bh, by, -ch, -bo }  { aj,ao, -ak, -ai, -an, al, ah, am, -am, -ah, -al, an, ai, ak, -ao, -aj, -aj,-ao, ak, ai, an, -al, -ah, -am, am, ah, al, -an, -ai, -ak, ao, aj, aj,ao, -ak, -ai, -an, al, ah, am, -am, -ah, -al, an, ai, ak, -ao, -aj, -aj,-ao, ak, ai, an, -al, -ah, -am, am, ah, al, -an, -ai, -ak, ao, aj }  {bp, ck, -bq, -bo, -cj, br, bn, ci, -bs, -bm, -ch, bt, bl, cg, -bu, -bk,-cf, by, bj, ce, -bw, -bi, -cd, bx, bh, cc, -by, -bg, -cb, bz, bf, ca,-ca, -bf, -bz, cb, bg, by, -cc, -bh, -bx, cd, bi, bw, -ce, -bj, -by, cf,bk, bu, -cg, -bl, -bt, ch, bm, bs, -ci, -bn, -br, cj, bo, bq, -ck, -bp } { au, -be, -at, -ay, bd, as, aw, -bc, -ar, -ax, bb, aq, ay, -ba, -ap,-az, az, ap, ba, -ay, -aq, -bb, ax, ar, bc, -aw, -as, -bd, ay, at, be,-au, -au, be, at, ay, -bd, -as, -aw, bc, ar, ax, -bb, -aq, -ay, ba, ap,az, -az, -ap, -ba, ay, aq, bb, -ax, -ar, -bc, aw, as, bd, -ay, -at, -be,au }  { bq, -ci, -bi, -by, cd, bg, ca, -by, -bi, -cf, bt, bn, ck, -bo,-bs, cg, bj, bx, -cb, -bf, -cc, bw, bk, ch, -br, -bp, cj, bm, bu, -ce,-bh, -bz, bz, bh, ce, -bu, -bm, -cj, bp, br, -ch, -bk, -bw, cc, bf, cb,-bx, -bj, -cg, bs, bo, -ck, -bn, -bt, cf, bi, by, -ca, -bg, -cd, by, bi,ci, -bq }  { ae, -ag, -ad, -af, af, ad, ag, -ae, -ae, ag, ad, af, -af,-ad, -ag, ae, ae, -ag, -ad, -af, af, ad, ag, -ae, -ae, ag, ad, af, -af,-ad, -ag, ae, ae, -ag, -ad, -af, af, ad, ag, -ae, -ae, ag, ad, af, -af,-ad, -ag, ae, ae, -ag, -ad, -af, af, ad, ag, -ae, -ae, ag, ad, af, -af,-ad, -ag, ae }  { br, -cf, -bg, -cc, bu, bo, -ci, -bj, -bz, bx, bi, ck,-bm, -bw, ca, bi, ch, -bp, -bt, cd, bf, ce, -bs, -bq, cg, bh, cb, -by,-bn, cj, bk, by, -by, -bk, -cj, bn, by, -cb, -bh, -cg, bq, bs, -se, -bf,-cd, bt, bp, -ch, -bi, -ca, bw, bm, -ck, -bi, -bx, bz, bj, ci, -bo, -bu,cc, bg, cf, -br }  { ay, -bb, -ap, -bc, au, aw, -ba, -aq, -bd, at, ax,-az, -ar, -be, as, ay, -ay, -as, be, ar, az, -ax, -at, bd, aq, ba, -aw,-au, bc, ap, bb, -ay, -ay, bb, ap, bc, -au, -aw, ba, aq, bd, -at, -ax,az, ar, be, -as, -ay, ay, as, -be, -ar, -az, ax, at, -bd, -aq, -ba, aw,au, -bc, -ap, -bb, ay }  { bs, -cc, -bi, -cj, bi, bz, -by, -bp, cf, bf,cg, -bo, -bw, by, bm, -ci, -bh, -cd, br, bt, -cb, -bj, -ck, bk, ca, -bu,-bq, ce, bg, ch, -bn, -bx, bx, bn, -ch, -bg, -ce, bq, bu, -ca, -bk, ck,bj, cb, -bt, -br, cd, bh, ci, -bm, -by, bw, bo, -cg, -bf, -cf, bp, by,-bz, -bi, cj, bi, cc, -bs }  { ak, -am, -ai, ao, ah, an, -aj, -al, al,aj, -an, -ah, -ao, ai, am, -ak, -ak, am, ai, -ao, -ah, -an, aj, al, -al,-aj, an, ah, ao, -ai, -am, ak, ak, -am, -ai, ao, ah, an, -aj, -al, al,aj, -an, -ah, -ao, ai, am, -ak, -ak, am, ai, -ao, -ah, -an, aj, al, -al,-aj, an, ah, ao, -ai, -am, ak }  { bt, -bz, -bn, cf, bh, ck, -bi, -ce,bo, by, -bu, -bs, ca, bm, -cg, -bg, -cj, bj, cd, -bp, -bx, by, br, -cb,-bl, ch, bf, ci, -bk, -cc, bq, bw, -bw, -bq, cc, bk, -ci, -bf, -ch, bl,cb, -br, -by, bx, bp, -cd, -bj, cj, bg, cg, -bm, -ca, bs, bu, -by, -bo,ce, bi, -ck, -bh, -cf, bn, bz, -bt }  { aw, -ay, -au, ba, as, -bc, -aq,be, ap, bd, -ar, -bb, at, az, -ay, -ax, ax, ay, -az, -at, bb, ar, -bd,-ap, -be, aq, bc, -as, -ba, au, ay, -aw, -aw, ay, au, -ba, -as, bc, aq,-be, -ap, -bd, ar, bb, -at, -az, ay, ax, -ax, -ay, az, at, -bb, -ar, bd,ap, be, -aq, -bc, as, ba, -au, -ay, aw }  { bu, -bw, -bs, by, bq, -ca,-bo, cc, bm, -ce, -bk, cg, bi, -ci, -bg, ck, bf, cj, -bh, -ch, bj, cf,-bi, -cd, bn, cb, -bp, -bz, br, bx, -bt, -by, by, bt, -bx, -br, bz, bp,-cb, -bn, cd, bi, -cf, -bj, ch, bh, -cj, -bf, -ck, bg, ci, -bi, -cg, bk,ce, -bm, -cc, bo, ca, -bq, -by, bs, bw, -bu }  { aa, -aa, -aa, aa, aa,-aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa,-aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa,-aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa,-aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa }  { by, -bt, -bx, br,bz, -bp, -cb, bn, cd, -bi, -cf, bj, ch, -bh, -cj, bf, -ck, -bg, ci, bi,-cg, -bk, ce, bm, -cc, -bo, ca, bq, -by, -bs, bw, bu, -bu, -bw, bs, by,-bq, -ca, bo, cc, -bm, -se, bk, cg, -bi, -ci, bg, ck, -bf, cj, bh, -ch,-bj, cf, bi, -cd, -bn, cb, bp, -bz, -br, bx, bt, -by }  { ax, -ay, -az,at, bb, -ar, -bd, ap, -be, -aq, bc, as, -ba, -au, ay, aw, -aw, -ay, au,ba, -as, -bc, aq, be, -ap, bd, ar, -bb, -at, az, ay, -ax, -ax, ay, az,-at, -bb, ar, bd, -ap, be, aq, -bc, -as, ba, au, -ay, -aw, aw, ay, -au,-ba, as, bc, -aq, -be, ap, -bd, -ar, bb, at, -az, -ay, ax }  { bw, -bq,-cc, bk, ci, -bf, ch, bi, -cb, -br, by, bx, -bp, -cd, bj, cj, -bg, cg,bm, -ca, -bs, bu, by, -bo, -ce, bi, ck, -bh, cf, bn, -bz, -bt, bt, bz,-bn, -cf, bh, -ck, -bi, ce, bo, -by, -bu, bs, ca, -bm, -cg, bg, -cj,-bj, cd, bp, -bx, -by, br, cb, -bi, -ch, bf, -ci, -bk, cc, bq, -bw }  {al, -aj, -an, ah, -ao, -ai, am, ak, -ak, -am, ai, ao, -ah, an, aj, -al,-al, aj, an, -ah, ao, ai, -am, -ak, ak, am, -ai, -ao, ah, -an, -aj, al,al, -aj, -an, ah, -ao, -ai, am, ak, -ak, -am, ai, ao, -ah, an, aj, -al,-al, aj, an, -ah, ao, ai, -am, -ak, ak, am, -ai, -ao, ah, -an, -aj, al } { bx, -bn, -ch, bg, -ce, -bq, bu, ca, -bk, -ck, bj, -cb, -bt, br, cd,-bh, ci, bm, -by, -bw, bo, cg, -bf, cf, bp, -by, -bz, bl, cj, -bi, cc,bs, -bs, -cc, bi, -cj, -bl, bz, by, -bp, -cf, bf, -cg, -bo, bw, by, -bm,-ci, bh, -cd, -br, bt, cb, -bj, ck, bk, -ca, -bu, bq, ce, -bg, ch, bn,-bx }  { ay, -as, -be, ar, -az, -ax, at, bd, -aq, ba, aw, -au, -bc, ap,-bb, -ay, ay, bb, -ap, bc, au, -aw, -ba, aq, -bd, -at, ax, az, -ar, be,as, -ay, -ay, as, be, -ar, az, ax, -at, -bd, aq, -ba, -aw, au, bc, -ap,bb, ay, -ay, -bb, ap, -bc, -au, aw, ba, -aq, bd, at, -ax, -az, ar, -be,-as, ay }  { by, -bk, cj, bn, -by, -cb, bh, -cg, -bq, bs, ce, -bf, cd,bt, -bp, -ch, bi, -ca, -bw, bm, ck, -bi, bx, bz, -bj, ci, bo, -bu, -cc,bg, -cf, -br, br, cf, -bg, cc, bu, -bo, -ci, bj, -bz, -bx, bi, -ck, -bm,bw, ca, -bi, ch, bp, -bt, -cd, bf, -ce, -bs, bq, cg, -bh, cb, by, -bn,-cj, bk, -by }  { af, -ad, ag, ae, -ae, -ag, ad, -af, -af, ad, -ag, -ae,ae, ag, -ad, af, af, -ad, ag, ae, -ae, -ag, ad, -af, -af, ad, -ag, -ae,ae, ag, -ad, af, af, -ad, ag, ae, -ae, -ag, ad, -af, -af, ad, -ag, -ae,ae, ag, -ad, af, af, -ad, ag, ae, -ae, -ag, ad, -af, -af, ad, -ag, -ae,ae, ag, -ad, af }  { bz, -bh, ce, bu, -bm, cj, bp, -br, -ch, bk, -bw,-cc, bf, -cb, -bx, bj, -cg, -bs, bo, ck, -bn, bt, cf, -bi, by, ca, -bg,cd, by, -bi, ci, bq, -bq, -ci, bi, -by, -cd, bg, -ca, -by, bi, -cf, -bt,bn, -ck, -bo, bs, cg, -bj, bx, cb, -bf, cc, bw, -bk, ch, br, -bp, -cj,bm, -bu, -ce, bh, -bz }  { az, -ap, ba, ay, -aq, bb, ax, -ar, bc, aw,-as, bd, ay, -at, be, au, -au, -be, at, -ay, -bd, as, -aw, -bc, ar, -ax,-bb, aq, -ay, -ba, ap, -az, -az, ap, -ba, -ay, aq, -bb, -ax, ar, -bc,-aw, as, -bd, -ay, at, -be, -au, au, be, -at, ay, bd, -as, aw, bc, -ar,ax, bb, -aq, ay, ba, -ap, az }  { ca, -bf, bz, cb, -bg, by, cc, -bh, bx,cd, -bi, bw, ce, -bj, by, cf, -bk, bu, cg, -bi, bt, ch, -bm, bs, ci,-bn, br, cj, -bo, bq, ck, -bp, bp, -ck, -bq, bo, -cj, -br, bn, -ci, -bs,bm, -ch, -bt, bi, -cg, -bu, bk, -cf, -by, bj, -se, -bw, bi, -cd, -bx,bh, -cc, -by, bg, -cb, -bz, bf, -ca }  { am, -ah, al, an, -ai, ak, ao,-aj, aj, -ao, -ak, ai, -an, -al, ah, -am, -am, ah, -al, -an, ai, -ak,-ao, aj, -aj, ao, ak, -ai, an, al, -ah, am, am, -ah, al, an, -ai, ak,ao, -aj, aj, -ao, -ak, ai, -an, -al, ah, -am, -am, ah, -al, -an, ai,-ak, -ao, aj, -aj, ao, ak, -ai, an, al, -ah, am }  { cb, -bi, bu, ci,-bp, bn, -cg, -bw, bg, -bz, -cd, bk, -bs, -ck, br, -bl, ce, by, -bf, bx,cf, -bm, bq, -cj, -bt, bj, -cc, -ca, bh, -by, -ch, bo, -bo, ch, by, -bh,ca, cc, -bj, bt, cj, -bq, bm, -cf, -bx, bf, -by, -ce, bl, -br, ck, bs,-bk, cd, bz, -bg, bw, cg, -bn, bp, -ci, -bu, bi, -cb }  { ba, -ar, ay,-be, -aw, aq, -az, -bb, as, -au, bd, ax, -ap, ay, bc, -at, at, -bc, -ay,ap, -ax, -bd, au, -as, bb, az, -aq, aw, be, -ay, ar, -ba, -ba, ar, -ay,be, aw, -aq, az, bb, -as, au, -bd, -ax, ap, -ay, -bc, at, -at, bc, ay,-ap, ax, bd, -au, as, -bb, -az, aq, -aw, -be, ay, -ar, ba }  { cc, -bi,bp, -cg, -by, bh, -bt, ck, bu, -bg, bx, ch, -bq, bk, -cb, -cd, bm, -bo,cf, bz, -bi, bs, -cj, -by, bf, -bw, -ci, br, -bj, ca, ce, -bn, bn, -ce,-ca, bj, -br, ci, bw, -bf, by, cj, -bs, bi, -bz, -cf, bo, -bm, cd, cb,-bk, bq, -ch, -bx, bg, -bu, -ck, bt, -bh, by, cg, -bp, bi, -cc }  { ac,-ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac,-ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac,-ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac,-ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac }  {cd, -bo, bk, -bz, -ch, bs, -bg, by, -ck, -bw, bh, -br, cg, ca, -bi, bn,-cc, -ce, bp, -bj, by, ci, -bt, bf, -bu, cj, bx, -bi, bq, -cf, -cb, bm,-bm, cb, cf, -bq, bi, -bx, -cj, bu, -bf, bt, -ci, -by, bj, -bp, ce, cc,-bn, bi, -ca, -cg, br, -bh, bw, ck, -by, bg, -bs, ch, bz, -bk, bo, -cd } { bb, -au, aq, -ax, be, ay, -ar, at, -ba, -bc, ay, -ap, aw, -bd, -az,as, -as, az, bd, -aw, ap, -ay, bc, ba, -at, ar, -ay, -be, ax, -aq, au,-bb, -bb, au, -aq, ax, -be, -ay, ar, -at, ba, bc, -ay, ap, -aw, bd, az,-as, as, -az, -bd, aw, -ap, ay, -bc, -ba, at, -ar, ay, be, -ax, aq, -au,bb }  { ce, -br, bf, -bs, cf, cd, -bq, bg, -bt, cg, cc, -bp, bh, -bu,ch, cb, -bo, bi, -by, ci, ca, -bn, bj, -bw, cj, bz, -bm, bk, -bx, ck,by, -bi, bi, -by, -ck, bx, -bk, bm, -bz, -cj, bw, -bj, bn, -ca, -ci, by,-bi, bo, -cb, -ch, bu, -bh, bp, -cc, -cg, bt, -bg, bq, -cd, -cf, bs,-bf, br, -ce }  { an, -ak, ah, -aj, am, ao, -al, ai, -ai, al, -ao, -am,aj, -ah, ak, -an, -an, ak, -ah, aj, -am, -ao, al, -ai, ai, -al, ao, am,-aj, ah, -ak, an, an, -ak, ah, -aj, am, ao, -al, ai, -ai, al, -ao, -am,aj, -ah, ak, -an, -an, ak, -ah, aj, -am, -ao, al, -ai, ai, -al, ao, am,-aj, ah, -ak, an }  { cf, -bu, bj, -bl, bw, -ch, -cd, bs, -bh, bn, -by,cj, cb, -bq, bf, -bp, ca, ck, -bz, bo, -bg, br, -cc, -ci, bx, -bm, bi,-bt, ce, cg, -by, bk, -bk, by, -cg, -ce, bt, -bi, bm, -bx, ci, cc, -br,bg, -bo, bz, -ck, -ca, bp, -bf, bq, -cb, -cj, by, -bn, bh, -bs, cd, ch,-bw, bl, -bj, bu, -cf }  { bc, -ax, as, -aq, ay, -ba, -be, az, -au, ap,-at, ay, -bd, -bb, aw, -ar, ar, -aw, bb, bd, -ay, at, -ap, au, -az, be,ba, -ay, aq, -as, ax, -bc, -bc, ax, -as, aq, -ay, ba, be, -az, au, -ap,at, -ay, bd, bb, -aw, ar, -ar, aw, -bb, -bd, ay, -at, ap, -au, az, -be,-ba, ay, -aq, as, -ax, bc }  { cg, -bx, bo, -bf, bn, -bw, cf, ch, -by,bp, -bg, bm, -by, ce, ci, -bz, bq, -bh, bi, -bu, cd, cj, -ca, br, -bi,bk, -bt, cc, ck, -cb, bs, -bj, bj, -bs, cb, -ck, -cc, bt, -bk, bi, -br,ca, -cj, -cd, bu, -bi, bh, -bq, bz, -ci, -ce, by, -bm, bg, -bp, by, -ch,-cf, bw, -bn, bf, -bo, bx, -cg }  { ag, -af, ae, -ad, ad, -ae, af, -ag,-ag, af, -ae, ad, -ad, ae, -af, ag, ag, -af, ae, -ad, ad, -ae, af, -ag,-ag, af, -ae, ad, -ad, ae, -af, ag, ag, -af, ae, -ad, ad, -ae, af, -ag,-ag, af, -ae, ad, -ad, ae, -af, ag, ag, -af, ae, -ad, ad, -ae, af, -ag,-ag, af, -ae, ad, -ad, ae, -af, ag }  { ch, -ca, bt, -bm, bf, -bi, bs,-bz, cg, ci, -cb, bu, -bn, bg, -bk, br, -by, cf, cj, -cc, by, -bo, bh,-bj, bq, -bx, ce, ck, -cd, bw, -bp, bi, -bi, bp, -bw, cd, -ck, -ce, bx,-bq, bj, -bh, bo, -by, cc, -cj, -cf, by, -br, bk, -bg, bn, -bu, cb, -ci,-cg, bz, -bs, bi, -bf, bm, -bt, ca, -ch }  { bd, -ba, ax, -au, ar, -ap,as, -ay, ay, -bb, be, bc, -az, aw, -at, aq, -aq, at, -aw, az, -bc, -be,bb, -ay, ay, -as, ap, -ar, au, -ax, ba, -bd, -bd, ba, -ax, au, -ar, ap,-as, ay, -ay, bb, -be, -bc, az, -aw, at, -aq, aq, -at, aw, -az, bc, be,-bb, ay, -ay, as, -ap, ar, -au, ax, -ba, bd }  { ci, -cd, by, -bt, bo,-bj, bf, -bk, bp, -bu, bz, -ce, cj, ch, -cc, bx, -bs, bn, -bi, bg, -bi,bq, -by, ca, -cf, ck, cg, -cb, bw, -br, bm, -bh, bh, -bm, br, -bw, cb,-cg, -ck, cf, -ca, by, -bq, bi, -bg, bi, -bn, bs, -bx, cc, -ch, -cj, ce,-bz, bu, -bp, bk, -bf, bj, -bo, bt, -by, cd, -ci }  { ao, -an, am, -al,ak, -aj, ai, -ah, ah, -ai, aj, -ak, al, -am, an, -ao, -ao, an, -am, al,-ak, aj, -ai, ah, -ah, ai, -aj, ak, -al, am, -an, ao, ao, -an, am, -al,ak, -aj, ai, -ah, ah, -ai, aj, -ak, al, -am, an, -ao, -ao, an, -am, al,-ak, aj, -ai, ah, -ah, ai, -aj, ak, -al, am, -an, ao }  { cj, -cg, cd,-ca, bx, -bu, br, -bo, bl, -bi, bf, -bh, bk, -bn, bq, -bt, bw, -bz, cc,-cf, ci, ck, -ch, ce, -cb, by, -by, bs, -bp, bm, -bj, bg, -bg, bj, -bm,bp, -bs, by, -by, cb, -ce, ch, -ck, -ci, cf, -cc, bz, -bw, bt, -bq, bn,-bk, bh, -bf, bi, -bl, bo, -br, bu, -bx, ca, -cd, cg, -cj }  { be, -bd,bc, -bb, ba, -az, ay, -ax, aw, -ay, au, -at, as, -ar, aq, -ap, ap, -aq,ar, -as, at, -au, ay, -aw, ax, -ay, az, -ba, bb, -bc, bd, -be, -be, bd,-bc, bb, -ba, az, -ay, ax, -aw, ay, -au, at, -as, ar, -aq, ap, -ap, aq,-ar, as, -at, au, -ay, aw, -ax, ay, -az, ba, -bb, bc, -bd, be }  { ck,-cj, ci, -ch, cg, -cf, ce, -cd, cc, -cb, ca, -bz, by, -bx, bw, -by, bu,-bt, bs, -br, bq, -bp, bo, -bn, bm, -bi, bk, -bj, bi, -bh, bg, -bf, bf,-bg, bh, -bi, bj, -bk, bi, -bm, bn, -bo, bp, -bq, br, -bs, bt, -bu, by,-bw, bx, -by, bz, -ca, cb, -cc, cd, -ce, cf, -cg, ch, -ci, cj, -ck } }where { aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap,aq, ar, as, at, au, ay, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh,bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, by, bw, bx, by, bz,ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck} = {64, 83, 36, 89, 75, 50,18, 90, 87, 80, 70, 57, 43, 25, 9, 90, 90, 88, 85, 82, 78, 73, 67, 61,54, 46, 38, 31, 22, 13, 4, 91, 90, 90, 90, 88, 87, 86, 84, 83, 81, 79,77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44, 41, 37, 33, 28, 24, 20, 15,11, 7, 2}

APPENDIX III 4-point DST-7 { a, b, c, d } { c, c, 0, -c } { d, -a, -c, b} { b, -d, c, -a } where {a, b, c, d} ={ 29, 55, 74, 84} 8-point DST-7:{ a, b, c, d, e, f, g, h,} { c, f, h, e, b, -a, -d, -g,} { e, g, b, -c,-h, -d, a, f,} { g, c, -d, -f, a, h, b, -e,} { h, -a, -g, b, f, -c, -e,d,} { f, -e, -a, g, -d, -b, h, -c,} { d, -h, e, -a, -c, g, -f, b,} { b,-d, f, -h, g, -e, c, -a,} where {a, b, c, d, e, f, g, h} = { 17, 32, 46,60, 71, 78, 85, 86} 16-point DST-7 { a, b, c, d, e, f, g, h, i, j, k, l,m, n, o, p,} { c, f, i, l, o, o, l, i, f, c, 0, -c, -f, -i, -l, -o,} {e, j, o, m, h, c, -b, -g, -l, -p, -k, -f, -a, d, i, n,} { g, n, l, e,-b, -i, -p, -j, -c, d, k, o, h, a, -f, -m,} { i, o, f, -c, -l, -l, -c,f, o, i, 0, -i, -o, -f, c, l,} { k, k, 0, -k, -k, 0, k, k, 0, -k, -k, 0,k, k, 0, -k,} { m, g, -f, -n, -a, l, h, -e, -o, -b, k, i, -d, -p, -c,j,} { o, c, -l, -f, i, i, -f, -l, c, o, 0, -o, -c, l, f, -i,} { p, -a,-o, b, n, -c, -m, d, l, -e, -k, f, j, -g, -i, h,} { n, -e, -i, j, d, -o,a, m, -f, -h, k, c, -p, b, l, -g,} { l, -i, -c, o, -f, -f, o, -c, -i, l,0, -l, i, c, -o, f,} { j, -m, c, g, -p, f, d, -n, i, a, -k, l, -b, -h,o, -e,} { h, -p, i, -a, -g, o, -j, b, f, -n, k, -c, -e, m, -l, d,} { f,-l, o, -i, c, c, -i, o, -l, f, 0, -f, l, -o, i, -c,} { d, -h, l, -p, m,-i, e, -a, -c, g, -k, o, -n, j, -f, b,} { b, -d, f, -h, j, -l, n, -p, o,-m, k, -i, g, -e, c, -a,} where {a, b, c, d, e, f, g, h, i, j, k, l, m,n, o, p} = { 9, 17, 25, 33, 41, 49, 56, 62, 66, 72, 77, 81, 83, 87, 89,90} 32-point DST-7 { a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q,r, s, t, u, v, w, x, y, z, A, B, C, D, E, F,} { c, f, i, l, o, r, u, x,A, D, F, C, z, w, t, q, n, k, h, e, b, -a, -d, -g, -j, -m, -p, -s, -v,-y, -B, -E,} { e, j, o, t, y, D, D, y, t, o, j, e, 0, -e, -j, -o, -t,-y, -D, -D, -y, -t, -o, -j, -e, 0, e, j, o, t, y, D,} { g, n, u, B, D,w, p, i, b, -e, -l, -s, -z, -F, -y, -r, -k, -d, c, j, q, x, E, A, t, m,f, -a, -h, -o, -v, -C,} { i, r, A, c, t, k, b, -g, -p, -y, -E, -v, -m,-d, e, n, w, F, x, o, f, -c, -l, -u, -D, -z, -q, -h, a, j, s, B,} { k,v, F, u, j, -a, -l, -w, -E, -t, -i, b, m, x, D, s, h, -c, -n, -y, -C,-r, -g, d, o, z, B, q, f, -e, -p, -A,} { m, z, z, m, 0, -m, -z, -z, -m,0, m, z, z, m, 0, -m, -z, -z, -m, 0, m, z, z, m, 0, -m, -z, -z, -m, 0,m, z,} { o, D, t, e, -j, -y, -y, -j, e, t, D, o, 0, -o, -D, -t, -e, j,y, y, j, -e, -t, -D, -o, 0, o, D, t, e, -j,-y,} { q, E, n, -c, -t, -B,-k, f, w, y, h, -i, -z, -v, -e, l, C, s, b, -o, -F, -p, a, r, D, m, -d,-u, -A, -j, g, x,} { s, A, h, -k, -D, -p, c, v, x, e, -n, -F, -m, f, y,u, b, -q, -C, -j, i, B, r, -a, -t, -z, -g, l, E, o, -d, -w,} { u, w, b,-s, -y, -d, q, A, f, -o, -c, -h, m, E, j, -k, -F, -l, i, D, n, -g, -B,-p, e, z, r, -c, -x, -t, a, v,} { w, s, -d, -A, -o, h, E, k, -l, -D, -g,p, z, c, -t, -v, a, x, r, -e, -B, -n, i, F, j, -m, -C, -f, q, y, b, -u,}{ y, o, -j, -D, -e, t, t, -e, -D, -j, o, y, 0, -y, -o, j, D, e, -t, -t,e, D, j, -o, -y, 0, y, o, -j, -D, -e, t,} { A, k, -p, -v, e, F, f, -u,-q, j, B, a, -z, -l, o, w, -d, -E, -g, t, r, -i, -C, -b, y, m, -n, -x,c, D, h, -s,} { c, g, -v, -n, o, u, -h, -B, a, D, f, -w, -m, p, t, -i,-A, b, E, e, -x, -l, q, s, -j, -z, c, F, d, -y, -k, r,} { E, c, -B, -f,y, i, -v, -l, s, o, -p, -r, m, u, -j, -x, g, A, -d, -D, a, F, b, -c, -e,z, h, -w, -k, t, n, -q,} { F, -a, -E, b, D, -c, -C, d, B, -e, -A, f, z,-g, -y, h, x, -i, -w, j, v, -k, -u, l, t, -m, -s, n, r, -o, -q, p,} { D,-e, -y, j, t, -o, -o, t, j, -y, -e, D, 0, -D, e, y, -j, -t, o, o, -t,-j, y, e, -D, 0, D, -e, -y, j, t, -o,} { B, -i, -s, r, j, -A, -a, c, -h,-t, q, k, -z, -b, D, -g, -u, p, l, -y, -c, E, -f, -v, o, m, -x, -d, F,-e, -w, n,} { z, -m, -m, z, 0, -z, m, m, -z, 0, z, -m, -m, z, 0, -z, m,m, -z, 0, z, -m, -m, z, 0, -z, m, m, -z, 0, z, -m,} { x, -q, -g, E, -j,-n, A, -c, -u, t, d, -B, m, k, -D, f, r, -w, -a, y, -p, -h, F, -i, -o,z, -b, -v, s, e, -C, l,} { v, -u, -a, w, -t, -b, x, -s, -c, y, -r, -d,z, -q, -e, A, -p, -f, B, -o, -g, C, -n, -h, D, -m, -i, E, -l, -j, F,-k,} { t, -y, e, o, -D, j, j, -D, o, e, -y, t, 0, -t, y, -e, -o, D, -j,-j, D, -o, -e, y, -t, 0, t, -y, e, o, -D, j,} { r, -c, k, g, -y, v, -d,-n, F, -o, -c, u, -z, h, j, -B, s, -a, -q, D, -l, -f, x, -w, e, m, -E,p, b, -t, A, -i,} { p, -F, q, -a, -o, E, -r, b, n, -D, s, -c, -m, C, -t,d, l, -B, u, -e, -k, A, -v, f, j, -z, w, -g, -i, y, -x, h,} { n, -B, w,-i, -e, s, -F, r, -d, -j, x, -A, m, a, -o, C, -v, h, f, -t, E, -q, c, k,-y, z, -l, -b, p, -D, u, -g,} { l, -x, c, -q, e, g, -s, E, -v, j, b, -n,z, -A, o, -c, -i, u, -F, t, -h, -d, p, -B, y, -m, a, k, -w, D, -r, f,} {j, -t, D, -y, o, -e, -e, o, -y, D, -t, j, 0, -j, t, -D, y, -o, e, e, -o,y, -D, t, -j, 0, j, -t, D, -y, o, -e,} { h, -p, x, -F, y, -q, i, -a, -g,o, -w, E, -z, r, -j, b, f, -n, v, -D, A, -s, k, -c, -e, m, -u, C, -B, t,-l, d,} { f, -l, r, -x, D, -C, w, -q, k, -e, -a, g, -m, s, -y, E, -B, v,-p, j, -d, -b, h, -n, t, -z, F, -A, u, -o, i, -c,} { d, -h, l, -p, t,-x, B, -F, C, -y, u, -q, m, -i, e, -a, -c, g, -k, o, -s, w, -A, E, -D,z, -v, r, -n, j, -f, b,} { b, -d, f, -h, j, -l, n, -p, r, -t, v, -x, z,-B, D, -F, E, -C, A, -y, w, -u, s, -q, o, -m, k, -i, g, -e, c, -a,}where {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v,w, x, y, z, A, B, C, D, E, F } = { 4, 9, 13, 17, 21, 26, 30, 34, 38, 42,45, 50, 53, 56, 60, 63, 66, 68, 72, 74, 77, 78, 80, 82, 84, 85, 86, 88,88, 89, 90, 90 } 4-point DCT-8 { a, b, c, d,} { b, 0, -b, -b,} { c, -b,-d, a,} { d, -b, a, -c,} where {a, b, c, d} = { 84, 74, 55, 29} 8-pointDCT-8: { a, b, c, d, e, f, g, h,} { b, e, h, -g, -d, -a, -c, -f,} { c,h, -e, -a, -f, g, b, d,} { d, -g, -a, -h, c, e, -f, -b,} { e, -d, -f, c,g, -b, -h, a,} { f, -a, g, e, -b, h, d, -c,} { g, -c, b, -f, -h, d, -a,e,} { h, -f, d, -b, a, -c, e, -g,} where {a, b, c, d, e, f, g, h} = {86, 85, 78, 71, 60, 46, 32, 17} 16-point DCT-8 { a, b, c, d, e, f, g, h,i, j, k, l, m, n, o, p,} { b, e, h, k, n, 0, -n, -k, -h, -e, -b, -b, -e,-h, -k, -n,} { c, h, m, -p, -k, -f, -a, -e, -j, -o, n, i, d, b, g, l,} {d, k, -p, -i, -b, -f, -m, n, g, a, h, o, -l, -e, -c, -j,} { e, n, -k,-b, -h, 0, h, b, k, -n, -e, -e, -n, k, b, h,} { f, 0, -f, -f, 0, f, f,0, -f, -f, 0, f, f, 0, -f, -f,} { g, -n, -a, -m, h, f, -o, -b, -l, i, e,-p, -c, -k, j, d,} { h, -k, -e, n, b, 0, -b, -n, e, k, -h, -h, k, e, -n,-b,} { i, -h, -j, g, k, -f, -l, e, m, -d, -n, c, o, -b, -p, a,} { j, -e,-o, a, -n, -f, i, k, -d, -p, b, -m, -g, h, l, -c,} { k, -b, n, h, -e, 0,e, -h, -n, b, -k, -k, b, -n, -h, e,} { l, -b, i, o, -e, f, -p, -h, c,-m, -k, a, -j, -n, d, -g,} { m, -e, d, -l, -n, f, -c, k, o, -g, b, -j,-p, h, -a, i,} { n, -h, b, -e, k, 0, -k, e, -b, h, -n, -n, h, -b, e,-k,} { o, -k, g, -c, b, -f, j, -n, -p, l, -h, d, -a, e, -i, m,} { p, -n,l, -j, h, -f, d, -b, a, -c, e, -g, i, -k, m, -o,} where {a, b, c, d, e,f, g, h, i, j, k, l, m, n, o, p} = { 90, 89, 87, 83, 81, 77, 72, 66, 62,56, 49, 41, 33, 25, 17, 9} 32-point DCT-8 { a, b, c, d, e, f, g, h, i,j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, F,} {b, e, h, k, n, q, t, w, z, C, F, -E, -B, -y, -v, -s, -p, -m, -j, -g, -d,-a, -c, -f, -i, -l, -o, -r, -u, -x, -A, -D,} { c, h, m, r, w, B, 0, -B,-w, -r, -m, -h, -c, -c, -h, -m, -r, -w, -B, 0, B, w, r, m, h, c, c, h,m, r, w, B,} { d, k, r, y, F, -A, -t, -m, -f, -b, -i, -p, -w, -D, C, v,o, h, a, g, n, u, B, -E, -x, -q, -j, -c, -e, -l, -s, -z,} { e, n, w, F,-y, -p, -g, -c, -l, -u, -D, A, r, i, a, j, s, B, -C, -t, -k, -b, -h, -q,-z, E, v, m, d, f, o, x,} { f, q, B, -A, -p, -e, -g, -r, -C, z, o, d, h,s, D, -y, -n, -c, -i, -t, -E, x, m, b, j, u, F, -w, -l, -a, -k, -v,} {g, t, 0, -t, -g, -g, -t, 0, t, g, g, t, 0, -t, -g, -g, -t, 0, t, g, g,t, 0, -t, -g, -g, -t, 0, t, g, g, t,} { h, w, -B, -m, -c, -r, 0, r, c,m, B, -w, -h, -h, -w, B, m, c, r, 0, -r, -c, -m, -B, w, h, h, w, -B, -m,-c, -r,} { i, z, -w, -f, -l, -C, t, c, o, F, -q, -a, -r, E, n, d, u, -B,-k, -g, -x, y, h, j, A, -v, -e, -m, -D, s, b, p,} { j, C, -r, -b, -u, z,g, m, F, -o, -e, -x, w, d, p, -E, -l, -h, -A, t, a, s, -B, -i, -k, -D,q, c, v, -y, -f, -n,} { k, F, -m, -i, -D, o, g, B, -q, -e, -z, s, c, x,-u, -a, -v, w, b, t, -y, -d, -r, A, f, p, -C, -h, -n, E, j, l,} { l, -E,-h, -p, A, d, t, -w, -a, -x, s, e, B, -o, -i, -F, k, m, -D, -g, -q, z,c, u, -v, -b, -y, r, f, C, -n, -j,} { m, -B, -c, -w, r, h, 0, -h, -r, w,c, B, -m, -m, B, c, w, -r, -h, 0, h, r, -w, -c, -B, m, m, -B, -c, -w, r,h,} { n, -y, -c, -D, i, s, -t, -h, E, d, x, -o, -m, z, b, C, -j, -r, u,g, -F, -e, -w, p, l, -A, -a, -B, k, q, -v, -f,} { o, -v, -h, C, a, D,-g, -w, n, p, -u, -i, B, b, E, -f, -x, m, q, -t, -j, A, c, F, -e, -y, l,r, -s, -k, z, d,} { p, -s, -m, v, j, -y, -g, B, d, -E, -a, -F, c, C, -f,-z, i, w, -l, -t, o, q, -r, -n, u, k, -x, -h, A, e, -D, -b,} { q, -p,-r, o, s, -n, -t, m, u, -l, -v, k, w, -j, -x, i, y, -h, -z, g, A, -f,-B, e, C, -d, -D, c, E, -b, -F, a,} { r, -m, -w, h, B, -c, 0, c, -B, -h,w, m, -r, -r, m, w, -h, -B, c, 0, -c, B, h, -w, -m, r, r, -m, -w, h, B,-c,} { s, -j, -B, a, -C, -i, t, r, -k, -A, b, -D, -h, u, q, -l, -z, c,-E, -g, v, p, -m, -y, d, -F, -f, w, o, -n, -x, e,} { t, -g, 0, g, -t,-t, g, 0, -g, t, t, -g, 0, g, -t, -t, g, 0, -g, t, t, -g, 0, g, -t, -t,g, 0, -g, t, t, -g, } { u, -d, B, n, -k, -E, g, -r, -x, a, -y, -q, h,-F, -j, o, A, -c, v, t, -e, C, m, -l, -D, f, -s, -w, b, -z, -p, i,} { v,-a, w, u, -b, x, t, -c, y, s, -d, z, r, -e, A, q, -f, B, p, -g, C, o,-h, D, n, -i, E, m, -j, F, l, -k,} { w, -c, r, B, -h, m, 0, -m, h, -B,-r, c, -w, -w, c, -r, -B, h, -m, 0, m, -h, B, r, -c, w, w, -c, r, B, -h,m,} { x, -f, m, -E, -q, b, -t, -B, j, -i, A, u, -c, p, F, -n, e, -w, -y,g, -l, D, r, -a, s, C, -k, h, -z, -v, d, -o,} { y, -i, h, -x, -z, j, -g,w, A, -k, f, -v, -B, l, -e, u, C, -m, d, -t, -D, n, -c, s, E, -o, b, -r,-F, p, -a, q,} { z, -l, c, -q, E, u, -g, h, -v, -D, p, -b, m, -A, -y, k,-d, r, -F, -t, f, -i, w, C, -o, a, -n, B, x, -j, e, -s,} { A, -o, c, -j,v, F, -t, h, -e, q, -C, -y, m, -a, l, -x, -D, r, -f, g, -s, E, w, -k, b,-n, z, B, -p, d, -i, u,} { B, -r, h, -c, m, -w, 0, w, -m, c, -h, r, -B,-B, r, -h, c, -m, w, 0, -w, m, -c, h, -r, B, B, -r, h, -c, m, -w,} { C,-u, m, -e, d, -l, t, -B, -D, v, -n, f, -c, k, -s, A, E, -w, o, -g, b,-j, r, -z, -F, x, -p, h, -a, i, -q, y,} { D, -x, r, -l, f, -a, g, -m, s,-y, E, C, -w, q, -k, e, -b, h, -n, t, -z, F, B, -v, p, -j, d, -c, i, -o,u, -A,} { E, -A, w, -s, o, -k, g, -c, b, -f, j, -n, r, -v, z, -D, -F, B,-x, t, -p, l, -h, d, -a, e, -i, m, -q, u, -y, C,} { F, -D, B, -z, x, -v,t, -r, p, -n, l, -j, h, -f, d, -b, a, -c, e, -g, i, -k, m, -o, q, -s, u,-w, y, -A, C, -E,} where {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o,p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, F } = {90, 90, 89, 88,88, 86, 85, 84, 82, 80, 78, 77, 74, 72, 68, 66, 63, 60, 56, 53, 50, 45,42, 38, 34, 30, 26, 21, 17, 13, 9, 4}

What is claimed is:
 1. A method for video decoding in a decoder,comprising: decoding coding information of a coefficient block from acoded video bitstream, the coding information indicating a size of thecoefficient block; determining, based on the size of the coefficientblock, whether to reduce a number of calculations in one of inversehorizontal and inverse vertical transforms of an inverse primarytransform, the inverse vertical transform transforming transformcoefficients of the coefficient block to intermediate data of anintermediate block, and the inverse horizontal transform transformingthe intermediate data to residual data in a residual block; performingthe inverse primary transform, wherein: when the number of calculationsin the inverse vertical transform is determined to be reduced, top 16rows of the intermediate data in the intermediate block are calculatedby the inverse vertical transform and the remaining intermediate data inthe intermediate block are zero; and when the number of calculations inthe inverse horizontal transform is determined to be reduced, left 16columns of the residual data in the residual block are calculated by theinverse horizontal transform and the remaining residual data in theresidual block are zero; and reconstructing a sample in the residualblock based on the residual data.
 2. The method of claim 1, wherein thesize of the coefficient block is 32×64; the one of the inversehorizontal and inverse vertical transforms is the inverse verticaltransform; the determining includes determining that the number ofcalculations in the inverse vertical transform is to be reduced when thesize of the coefficient block is 32×64; and the performing the inverseprimary transform includes performing the inverse primary transform, thetop 16 rows of the intermediate data in the intermediate block beingcalculated by the inverse vertical transform and the remainingintermediate data in the intermediate block being zero.
 3. The method ofclaim 1, wherein the size of the coefficient block is 32×64; the one ofthe inverse horizontal and inverse vertical transforms is the inversehorizontal transform; the determining includes determining that thenumber of calculations in the inverse horizontal transform is to bereduced when the size of the coefficient block is 32×64; and theperforming the inverse primary transform includes performing the inverseprimary transform, the left 16 columns of the residual data in theresidual block being calculated by the inverse horizontal transform andthe remaining residual data in the residual block being zero.
 4. Themethod of claim 1, wherein: the size of the coefficient block is 32×32;the one of the inverse horizontal and inverse vertical transforms is theinverse horizontal transform; the determining includes determining thatthe number of calculations in the inverse horizontal transform is to bereduced when the size of the coefficient block is 32×32; and theperforming the inverse primary transform includes performing the inverseprimary transform, the left 16 columns of the residual data in theresidual block being calculated by the inverse horizontal transform, theremaining residual data in the residual block being zero, and theintermediate data in the intermediate block being calculated by theinverse vertical transform.
 5. The method of claim 1, wherein the sizeof the coefficient block is 32×32; the one of the inverse horizontal andinverse vertical transforms is the inverse vertical transform; thedetermining includes determining that the number of calculations in theinverse vertical transform is to be reduced when the size of thecoefficient block is 32×32; and the performing the inverse primarytransform includes performing the inverse primary transform, the top 16rows of the intermediate data in the intermediate block being calculatedby the inverse vertical transform, the remaining intermediate data inthe intermediate block being zero, and the residual data in the residualblock being calculated by the inverse horizontal transform.
 6. Anapparatus for video decoding, comprising processing circuitry configuredto: decode coding information of a coefficient block from a coded videobitstream, the coding information indicating a size of the coefficientblock; determine, based on the size of the coefficient block, whether toreduce a number of calculations in one of inverse horizontal and inversevertical transforms of an inverse primary transform, the inversevertical transform transforming transform coefficients of thecoefficient block to intermediate data of an intermediate block, and theinverse horizontal transform transforming the intermediate data toresidual data in a residual block; perform the inverse primarytransform, wherein: when the number of calculations in the inversevertical transform is determined to be reduced, top 16 rows of theintermediate data in the intermediate block are calculated by theinverse vertical transform and the remaining intermediate data in theintermediate block are zero; and when the number of calculations in theinverse horizontal transform is determined to be reduced, left 16columns of the residual data in the residual block are calculated by theinverse horizontal transform and the remaining residual data in theresidual block are zero; and reconstruct a sample in the residual blockbased on the residual data.
 7. The apparatus of claim 6, wherein thesize of the coefficient block is 32×64; the one of the inversehorizontal and inverse vertical transforms is the inverse verticaltransform; and the processing circuitry is further configured to:determine that the number of calculations in the inverse verticaltransform is to be reduced when the size of the coefficient block is32×64; and perform the inverse primary transform, wherein the top 16rows of the intermediate data in the intermediate block are calculatedby the inverse vertical transform and the remaining intermediate data inthe intermediate block are zero.
 8. The apparatus of claim 6, whereinthe size of the coefficient block is 32×64; the one of the inversehorizontal and inverse vertical transforms is the inverse horizontaltransform; and the processing circuitry is further configured to:determine that the number of calculations in the inverse horizontaltransform is to be reduced when the size of the coefficient block is32×64; and perform the inverse primary transform, wherein the left 16columns of the residual data in the residual block are calculated by theinverse horizontal transform and the remaining residual data in theresidual block are zero.
 9. The apparatus of claim 6, wherein: the sizeof the coefficient block is 32×32; the one of the inverse horizontal andinverse vertical transforms is the inverse horizontal transform; and theprocessing circuitry is further configured to: determine that the numberof calculations in the inverse horizontal transform is to be reducedwhen the size of the coefficient block is 32×32; and perform the inverseprimary transform, wherein the left 16 columns of the residual data inthe residual block are calculated by the inverse horizontal transform,the remaining residual data in the residual block are zero, and theintermediate data in the intermediate block are calculated by theinverse vertical transform.
 10. The apparatus of claim 6, wherein thesize of the coefficient block is 32×32; the one of the inversehorizontal and inverse vertical transforms is the inverse verticaltransform; and the processing circuitry is further configured to:determine that the number of calculations in the inverse verticaltransform is to be reduced when the size of the coefficient block is32×32; and perform the inverse primary transform, wherein the top 16rows of the intermediate data in the intermediate block are calculatedby the inverse vertical transform, the remaining intermediate data inthe intermediate block are zero, and the residual data in the residualblock are calculated by the inverse horizontal transform.